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+# How Powerful are $K$ -hop Message Passing Graph Neural Networks
+
+Jiarui Feng1,2 Yixin Chen1 Fuhai $\mathbf { L i } ^ { 2 }$ Anindya Sarkar1 Muhan Zhang3,4 {feng.jiarui, fuhai.li, anindya}@wustl.edu, chen@cse.wustl.edu, muhan@pku.edu.cn 1Department of CSE, Washington University in St. Louis 2Institute for Informatics, Washington University School of Medicine 3Institute for Artificial Intelligence, Peking University 4Beijing Institute for General Artificial Intelligence
+
+# Abstract
+
+The most popular design paradigm for Graph Neural Networks (GNNs) is 1-hop message passing—aggregating information from 1-hop neighbors repeatedly. However, the expressive power of 1-hop message passing is bounded by the WeisfeilerLehman (1-WL) test. Recently, researchers extended 1-hop message passing to $K$ -hop message passing by aggregating information from $K$ -hop neighbors of nodes simultaneously. However, there is no work on analyzing the expressive power of $K$ -hop message passing. In this work, we theoretically characterize the expressive power of $K$ -hop message passing. Specifically, we first formally differentiate two different kernels of $K$ -hop message passing which are often misused in previous works. We then characterize the expressive power of $K$ -hop message passing by showing that it is more powerful than 1-WL and can distinguish almost all regular graphs. Despite the higher expressive power, we show that $K$ -hop message passing still cannot distinguish some simple regular graphs and its expressive power is bounded by 3-WL. To further enhance its expressive power, we introduce a KP-GNN framework, which improves $K$ -hop message passing by leveraging the peripheral subgraph information in each hop. We show that KP-GNN can distinguish many distance regular graphs which could not be distinguished by previous distance encoding or 3-WL methods. Experimental results verify the expressive power and effectiveness of KP-GNN. KP-GNN achieves competitive results across all benchmark datasets.
+
+# 1 Introduction
+
+Currently, most existing graph neural networks (GNNs) follow the message passing framework, which iteratively aggregates information from the neighbors and updates the representations of nodes. It has shown superior performance on graph-related tasks [1, 2, 3, 4, 5, 6, 7] comparing to traditional graph embedding techniques [8, 9]. However, as the procedure of message passing is similar to the 1-dimensional Weisfeiler-Lehman (1-WL) test [10], the expressive power of message passing GNNs is also bounded by the 1-WL test [7, 11]. Namely, GNNs cannot distinguish two non-isomorphic graph structures if the 1-WL test fails.
+
+In normal message passing GNNs, the node representation is updated by the direct neighbors of the node, which are called 1-hop neighbors. Recently, some works extend the notion of message passing into $K$ -hop message passing [12, 13, 14, 15, 16]. $K$ -hop message passing is a type of message passing where the node representation is updated by aggregating information from not only 1st hop but all the neighbors within $K$ hops of the node. However, there is no work on theoretically characterizing the expressive power of GNNs with $K$ -hop message passing, e.g., whether it can improve the 1-hop message passing or not and to what extent it can.
+
+In this work, we theoretically characterize the expressive power of $K$ -hop message passing GNNs. Specifically, 1) we formally distinguish two different kernels of the $K$ -hop neighbors, which are often misused in previous works. The first kernel is based on whether the node can be reached within $k$ steps of the graph diffusion process, which is used in GPR-GNN [15] and MixHop [12]. The second one is based on the shortest path distance of $k$ , which is used in $\mathrm { G I N E + }$ [16] and Graphormer [17]. Further, we show that different kernels of $K$ -hop neighbors will result in different expressive power of $K$ -hop message passing. 2) We show that $K$ -hop message passing is strictly more powerful than 1-hop message passing and can distinguish almost all regular graphs. 3) However, it still failed in distinguishing some simple regular graphs, no matter which kernel is used, and its expressive power is bounded by 3-WL. This motivates us to improve $K$ -hop message passing further.
+
+Here, we introduce KP-GNN, a new GNN framework with $K$ -hop message passing, which significantly improves the expressive power of standard $K$ -hop message passing GNNs. In particular, during the aggregation of neighbors in each hop, KP-GNN not only aggregates neighboring nodes in that hop but also aggregates the peripheral subgraph (subgraph induced by the neighbors in that hop). This additional information helps the KP-GNN to learn more expressive local structural features around the node. We further show that KP-GNN can distinguish many distance regular graphs with a proper encoder for the peripheral subgraph. The proposed KP-GNN has several additional advantages. First, it can be applied to most existing $K$ -hop message-passing GNNs with only slight modification. Second, it only adds little computational complexity to standard $K$ -hop message passing. We demonstrate the effectiveness of the KP-GNN framework through extensive experiments on both simulation and real-world datasets.
+
+# 2 $K$ -hop message passing and its expressive power
+
+# 2.1 Notations
+
+Denote a graph as $G = ( V , E )$ , where $V = \{ 1 , 2 , . . . , n \}$ is the node set and $E \subseteq V \times V$ is the edge set. Meanwhile, denote $\dot { A } \in \{ 0 , 1 \} ^ { n \times n }$ as the adjacency matrix of graph $G$ . Denote $x _ { v }$ as the feature vector of node $v$ and denote $e _ { u v }$ as the feature vector of the edge from $u$ to $v$ . Finally, we denote $Q _ { v , G } ^ { 1 }$ as the set of 1-hop neighbors of node $v$ in graph $G$ and $\mathcal { N } _ { v , G } ^ { 1 ^ { - } } { = } Q _ { v , G } ^ { 1 } \cup \{ v \}$ . Note that when we say $K$ -hop neighbors of node $v$ , we mean all the neighbors that have distance from node $v$ less than or equal to $K$ . In contrast, $k$ -th hop neighbors mean the neighbors with exactly distance $k$ from node $v$ . The definition of distance will be discussed in section 2.3.
+
+# 2.2 1-hop message passing framework
+
+Currently, most existing GNNs are designed based on 1-hop message passing framework [18]. Denote $h _ { v } ^ { l }$ as the output representation of node $v$ at layer $l$ and $h _ { v } ^ { 0 } = x _ { v }$ . Briefly, given a graph $G$ and a 1-hop message passing GNN, at layer $l$ of the GNN, $h _ { v } ^ { l }$ is computed by $h _ { v } ^ { l - 1 }$ and $\{ h _ { u } ^ { l - 1 } \mid u \in Q _ { v , G } ^ { 1 } \}$ :
+
+$$
+m _ { v } ^ { l } = \mathrm { { \bf M E S } } ^ { l } ( \{ ( h _ { u } ^ { l - 1 } , e _ { u v } ) | u \in Q _ { v , G } ^ { 1 } \} ) , h _ { v } ^ { l } = \mathrm { { \bf U P D } } ^ { l } ( m _ { v } ^ { l } , h _ { v } ^ { l - 1 } ) ,
+$$
+
+where $m _ { v } ^ { l }$ is the message to node $v$ at layer $l$ , $\mathrm { M E S } ^ { l }$ and $\mathrm { U P D } ^ { l }$ are message and update functions at layer $l$ respectively. After $L$ layers of message passing, $h _ { v } ^ { L }$ is used as the final representation of node $v$ . Such a representation can be used to conduct node-level tasks like node classification and node regression. To get the graph representation, a readout function is used:
+
+$$
+h _ { G } = \mathrm { R E A D O U T } ( \{ h _ { v } ^ { L } | v \in V \} ) ,
+$$
+
+where READOUT is the readout function for computing the final graph representation. Then $h _ { G }$ can be used to conduct graph-level tasks like graph classification and graph regression.
+
+# 2.3 $K$ -hop message passing framework
+
+The 1-hop message passing framework can be directly generalized to $K$ -hop message passing, as it shares the same message and update mechanism. The difference is that independent message and update functions can be employed for each hop. Meanwhile, a combination function is needed to combine the results from different hops into the final node representation at this layer. First, we differentiate two different kernels of $K$ -hop neighbors, which are interchanged and misused in previous research.
+
+The first kernel of $K$ -hop neighbors is shortest path distance (spd) kernel. Namely, the $k$ -th hop neighbors of node $v$ in graph $G$ is the set of nodes with the shortest path distance of $k$ from $v$ .
+
+Definition 1. For a node v in graph G, the K-hop neighbors N K,spdv,G of $v$ based on shortest path distance kernel is the set of nodes that have the shortest path distance from node $v$ less than or equal to $K$ . We further denote $Q _ { v , G } ^ { k , s p d }$ as the set of nodes in $G$ that are exactly the $k$ -th hop neighbors (with shortest path distance of exactly $k$ ) and $\mathcal { N } _ { v , G } ^ { 0 , s p d } = Q _ { v , G } ^ { 0 , s p d } = \{ v \}$ is the node itself.
+
+The second kernel of the $K$ -hop neighbors is based on graph diffusion $( g d )$ .
+
+Definition 2. For a node v in graph $G$ , the $K$ -hop neighbors $\mathcal { N } _ { v , G } ^ { K , g d }$ of v based on graph diffusion kernel is the sediffusion steps nodes that can diffuse iand the diffusion kernel ormation to node (adjacency matr $v$ within the number). We further denote dom walk as the set $K$ $A$ $Q _ { v , G } ^ { k , g d }$ of nodes in that are exactly the -th hop neighbors (nodes that can diffuse information to node with k diffusion steps) and N 0,gdv,G $\mathcal { N } _ { v , G } ^ { 0 , g d } = Q _ { v , G } ^ { 0 , g \bar { d } } = \bar { \{ v \} }$ is the node itself.
+
+Note that a node can be a $k$ -th hop neighbor of $v$ for multiple $k$ based on the graph diffusion kernel, but it can only appear in one hop for the shortest path distance kernel. We include more discussions of $K$ -hop kernels in Appendix A. Next, we define the $K$ -hop message passing framework as follows:
+
+$$
+\begin{array} { r } { m _ { v } ^ { l , k } = \mathrm { M E S } _ { k } ^ { l } ( \{ ( h _ { u } ^ { l - 1 } , e _ { u v } ) | u \in Q _ { v , G } ^ { k , t } ) \} ) , h _ { v } ^ { l , k } = \mathrm { U P D } _ { k } ^ { l } ( m _ { v } ^ { l , k } , h _ { v } ^ { l - 1 } ) , } \\ { h _ { v } ^ { l } = \mathrm { C O M B I N E } ^ { l } ( \{ \{ h _ { v } ^ { l , k } | k = 1 , 2 , . . . , K \} \} ) , } \end{array}
+$$
+
+where $t \in \{ s p d , g d \}$ , spd is the shortest path distance kernel and $_ { g d }$ is the graph diffusion kernel. Here, for each hop, we can apply unique MES and UPD functions. Note that for $k > 1$ , there may not exist the edge feature $e _ { u v }$ as nodes are not directly connected. But we leave it here since we can use other types of features to replace it like path encoding. We further discuss it in Appendix I. Compared to the 1-hop message passing framework described in Equation (1), the COMBINE function is introduced to combine the representations of node $v$ at different hops. It is easy to see that a $L$ layer 1-hop message passing GNNs is actually a $L$ layer $K$ -hop message passing GNNs with $K = 1$ . We include more discussions of $K$ -hop message passing GNNs in Appendix A. To aid further analysis, we also prove that $K$ -hop message passing can injectively encode the neighbor representations at different hops into $h _ { v } ^ { l }$ in Appendix B.
+
+# 2.4 Expressive power of $K$ -hop message passing framework
+
+In this section, we theoretically analyze the expressive power of $K$ -hop message passing. We assume there is no edge feature and all nodes in the graph have the same feature, which means that GNNs can only distinguish two nodes with the local structure of nodes. Note that including node features only increases the expressive power of GNNs as nodes/graphs are more easily to be discriminated. It has been proved that the expressive power of 1-hop message passing is bounded by the 1-WL test on discriminating non-isomorphic graphs [7, 11]. In this section, We show that the $K$ -hop message passing is strictly more powerful than the 1-WL test when $K > 1$ . Across the analysis, we utilize regular graphs as examples to illustrate our theorems since they cannot be distinguished using either 1-hop message passing or the 1-WL test. To begin the analysis, we first define proper $K$ -hop message passing GNNs.
+
+Definition 3. A proper $K$ -hop message passing GNN is a GNN model where the message, update, and combine functions are all injective given the input from a countable space.
+
+A proper $K$ -hop message passing GNN is easy to find due to the universal approximation theorem [19] of neural network and the Deep Set for set operation [20]. In the latter sections, by default, all mentioned $K$ -hop message passing GNNs are proper. Next, we introduce node configuration:
+
+Definition 4. The node configuration of node v in graph $G$ within $K$ hops under t kernel is a list $A _ { v , G } ^ { K , t } = ( a _ { v , G } ^ { 1 , t } , a _ { v , G } ^ { 2 , t } , . . . , a _ { v , G } ^ { K , t } )$ , where $a _ { v , G } ^ { i , t } = | Q _ { v , G } ^ { i , t } |$ is the number of $i$ -th hop neighbors of node v.
+
+![](images/6717718d9a0f12e65c4bf76a241bcd84384ca02b31ebd9cd9710377c107a2563.jpg)  
+Figure 1: Here are two pairs of non-isomorphic regular graphs. With 2-hop message passing, example 1 can be distinguished by the graph diffusion kernel, and example 2 can be distinguished by the shortest path distance kernel. However, both two examples become indistinguishable if we switch the kernel. Finally, both two examples can be distinguished by adding peripheral subgraph information.
+
+When we say two node configurations AK,t v1,G(1) and AK,t $A _ { v _ { 2 } , G ^ { ( 2 ) } } ^ { K , t }$ are equal, we mean that these two lists are component-wise equal to each other. Now, we state the first proposition:
+
+Proposition 1. A proper $K$ -hop message passing GNN is strictly more powerful than 1-hop message passing GNNs when $K > 1$ .
+
+To see why this is true, we first discuss how node configuration relates to the first layer of message passing. In the first layer of $K$ -hop message passing, each node aggregates neighbors from up to $K$ hops. As each node has the same node label, an injective message function can only know how many neighbors at each hop, which is exactly the node configuration. In other words, The first layer of $K$ -hop message passing is equivalent to inject node configuration to each node label. When $K = 1$ , the node configuration of $v _ { 1 }$ and $v _ { 2 }$ are $d _ { v _ { 1 } , G ^ { ( 1 ) } }$ and $d _ { v _ { 2 } , G ^ { ( 2 ) } }$ , where $d _ { v , G }$ is the node degree of $v$ . After $L$ layers, GNNs can only get the node degree information of each node within $L$ hops of node $v$ . Then, it is straightforward to see why these GNNs cannot distinguish any $n$ -sized $r$ -regular graph, as each node in the regular graph has the same degree.
+
+Next, when $K > 1$ , the $K$ -hop message passing is at least equally powerful as 1-hop message passing since node configuration up to $K$ hop includes all the information the 1 hop has. To see why it is more powerful, we use two examples to illustrate it. The first example is shown in the left part of Figure 1. Suppose here we use graph diffusion kernel and we want to learn the representation of node $v _ { 1 }$ and node $v _ { 2 }$ in the two graphs. We know that the 1-hop GNNs produce the same representation for two nodes as they are both nodes in 6-sized 3-regular graphs. However, it is easy to see that $v _ { 1 }$ and $v _ { 2 }$ have different local structures and should have different representations. Instead, if we use the 2-hop message passing with the graph diffusion kernel, we can easily distinguish two nodes by checking the 2nd hop neighbors of the node, as node $v _ { 1 }$ has four 2nd hop neighbors but node $v _ { 2 }$ only has two 2nd hop neighbors. The second example is shown in the right part of Figure 1. Two graphs in the example are still regular graphs. Suppose here we use shortest path distance kernel, node $v _ { 1 }$ and $v _ { 2 }$ have different numbers of 2nd hop neighbors and thus will have different representations by performing 2-hop message passing. These two examples convincingly demonstrate that the $K$ -hop message passing with $K > 1$ can have better expressive power than $K = 1$ . To further study the expressive power of $K$ -hop message passing on regular graphs, we show the following result:
+
+Theorem 1. Consider all pairs of $n$ -sized $r$ -regular graphs, let $3 \leq r < ( 2 l o g 2 n ) ^ { 1 / 2 }$ and $\epsilon$ be a fixed constant. With at most $\begin{array} { r } { K = \lfloor ( \frac { 1 } { 2 } + \epsilon ) \frac { \log { 2 n } } { \log { ( r - 1 ) } } \rfloor } \end{array}$ , there exists a $I$ layer $K$ -hop message passing GNN using the shortest path distance kernel that distinguishes almost all $1 - o ( n ^ { - 1 / 2 } )$ such pairs of graphs.
+
+We include the proof and simulation results in Appendix C. Theorem 1 shows that even with 1 layer and a modest $K$ , $K$ -hop GNNs are powerful enough to distinguish almost all regular graphs.
+
+Finally, we characterize the existing $K$ -hop methods with the proposed $K$ -hop message passing framework. Specifically, we show that 1) the expressive power of $K$ layer GINE [16] is bounded by $K$ layer $K$ -hop message passing with the shortest path distance kernel. 2) The expressive power of the Graphormer [17] is equal to $K$ -hop GNNs with the shortest path distance kernel and infinity $K$ . 3) For spectral GNNs and existing $K$ -hop GNNs with the graph diffusion kernel like MixHop [12] and MAGNA [14], we find they actually use a weak version of $K$ -hop than the definition of us. Specifically, it is shown that the expressive power of spectral GNNs is also bounded by 1-WL test [21], which contradicts our result as graph diffusion can be viewed as a special case of spectral GNN. However, we show that our definition of $K$ -hop message passing with graph diffusion kernel actually injects a non-linear function on the spectral basis, thus achieving superior expressive power. We leave the detailed discussion in Appendix D. Further, Distance Encoding [22] also uses the shortest path distance information to augment the 1-hop message passing, which is similar to $K$ -hop GNNs with the shortest path distance kernel. However, we find the expressive power of the two frameworks differs from each other. We leave the detailed discussion in Appendix E.
+
+# 2.5 Limitation of $K$ -hop message passing framework
+
+Although we show that $K$ -hop GNNs with $K > 1$ are better at distinguishing non-isomorphic structures than 1-hop GNNs, there are still limitations. In this section, we discuss the limitation of $K$ -hop message passing. Specifically, we show that the choice of the kernel can affect the expressive power of $K$ -hop message passing. Furthermore, even with $K$ -hop message passing, we still cannot distinguish some simple non-isomorphic structures and the expressive power of $K$ -hop message passing is bounded by 3-WL.
+
+Continue looking at the provided examples in Figure 1. In example 1, if we use the shortest path distance kernel instead of the graph diffusion kernel, two nodes have the same number of neighbors in the 2nd hop, which means that we cannot distinguish two nodes this time. Similarly, in example 2, two nodes have the same number of neighbors in both 1st and 2nd hops using graph diffusion kernel. These results highlight that the choice of the kernel can affect the expressive power of $K$ -hop message passing, and none of them can distinguish both two examples with 2-hop message passing.
+
+Recently, Frasca et al. [23] show that any subgraph-based GNNs with node-based selection policy can be implemented by 3-IGN [24, 25] and thus their expressive power is bounded by 3-WL test. Here, we show that the $K$ -hop message passing GNNs can also be implemented by 3-IGN for both two kernels and thus:
+
+Theorem 2. The expressive power of a proper $K$ -hop message passing GNN of any kernel is bounded by the 3-WL test.
+
+We include the proof in Appendix F. Given all these observations, we may wonder if there is a way to further improve the expressive power of $K$ -hop message passing?
+
+# 3 KP-GNN: improving the power of $K$ -hop message passing by peripheral subgraph
+
+In this section, we describe how to improve the expressive power of $K$ -hop message passing by adding additional information to the message passing framework. Specifically, by adding peripheral subgraph information, we can improve the expressive power of the $K$ -hop message passing by a large margin.
+
+# 3.1 Peripheral edge and peripheral subgraph
+
+First, we define peripheral edge and peripheral subgraph.
+
+Definition 5. The peripheral edge $E ( Q _ { v , G } ^ { k , t } )$ is defined as the set of edges that connect nodes within set $Q _ { v , G } ^ { k , t }$ . We further denote $| E ( Q _ { v , G } ^ { k , t } ) |$ as the number of peripheral edge in $E ( Q _ { v , G } ^ { k , t } )$ . The peripheral subgraph Gk,tv,G $G _ { v , G } ^ { k , t } = ( Q _ { v , G } ^ { k , t } , E ( Q _ { v , G } ^ { k , t } ) )$ is defined as the subgraph induced by $Q _ { v , G } ^ { k , t }$ from the whole graph $G$ .
+
+Briefly speaking, the peripheral edge $E ( Q _ { v , G } ^ { k , t } )$ record all the edges whose two ends are both from $Q _ { v , G } ^ { k , t }$ and the peripheral subgraph is a graph constituted by peripheral edges. It is easy to see that the peripheral subgraph $G _ { v , G } ^ { k , t }$ automatically contains all the information of peripheral edge $E ( Q _ { v , G } ^ { k , t } )$ . Next, we show that the power of $K$ -hop message passing can be improved by leveraging the information of peripheral edges and peripheral subgraphs. We again refer to the examples in Figure 1. Here we only consider the peripheral edge information. In example 1, we notice that at the 1st hop, there is an edge between node 3 and node 4 in the left graph. More specifically, $E ( Q _ { v _ { 1 } , G ^ { ( 1 ) } } ^ { 1 , t } ) \stackrel { - } { = } \{ ( 3 , 4 ) \}$ . In contrast, we have $E ( Q _ { v _ { 2 } , G ^ { ( 2 ) } } ^ { 1 , t } ) = \{ \}$ in the right graph, which means there is no edge between the 1st hop neighbors of $v _ { 2 }$ . Therefore, we can successfully distinguish these two nodes by adding this information to the message passing. Similarly, in example 2, there is one edge between the 1st hop neighbors of node $v _ { 2 }$ , but no such edge exists for node $v _ { 1 }$ . By leveraging peripheral edge information, we can also distinguish the two nodes. The above examples demonstrate the effectiveness of the peripheral edge and peripheral subgraph information.
+
+# 3.2 $K$ -hop peripheral-subgraph-enhanced graph neural network
+
+In this section, we propose $\mathbf { K }$ -hop Peripheral-subgraph-enhanced Graph Neural Network (KP-GNN), which equips $K$ -hop message passing GNNs with peripheral subgraph information for more powerful GNN design. Recall the $K$ -hop message passing defined in Equation (3). The only difference between KP-GNN and original $K$ -hop GNNs is that we revise the message function as follows:
+
+$$
+\begin{array} { r } { m _ { v } ^ { l , k } = \mathbf { M E S } _ { k } ^ { l } ( \{ ( h _ { u } ^ { l - 1 } , \ e _ { u v } ) | u \in Q _ { v , G } ^ { k , t } \} , \ G _ { v , G } ^ { k , t } ) . } \end{array}
+$$
+
+Briefly speaking, in the message step at the $k$ -th hop, we not only aggregate information of the neighbors but also the peripheral subgraph at that hop. The implementation of KP-GNN can be very flexible, as any graph encoding function can be used. To maximize the information the model can encode while keeping it simple, we implement the message function as:
+
+$$
+\begin{array} { r l } & { \mathbf { M E S } _ { k } ^ { l } = \mathbf { M E S } _ { k } ^ { l , n o r m a l } ( \ P ( h _ { u } ^ { l - 1 } , e _ { u v } ) | u \in Q _ { v , G } ^ { k , t } \| ) + f ( G _ { v , G } ^ { k , t } ) , } \\ & { \qquad f ( G _ { v , G } ^ { k , t } ) = \mathbf { E M B } ( ( E ( Q _ { v , G } ^ { k , t } ) , C _ { k } ^ { k ^ { \prime } } ) ) ~ , } \end{array}
+$$
+
+where MESl,normal denotes the message function in the original GNN model, $C _ { k } ^ { k ^ { \prime } }$ is the $k ^ { \prime }$ configuration, which encode both node configuration and the number of the peripheral edge of all nodes in $G _ { v , G } ^ { k , t }$ up to $k ^ { \prime }$ hops. It can be regarded as running another 1 layer KP-GNN and readout function on each peripheral subgraph. EMB is a learnable embedding function. With this implementation, any base GNN model can be incorporated into and be enhanced by the KP-GNN framework by replacing $\mathbf { M E S } _ { k } ^ { l , n o r m a l }$ and $\mathrm { U P D } _ { k } ^ { l }$ with the corresponding functions for each hop $k$ . We leave the detailed implementation in Appendix I.
+
+# 3.3 The expressive power of KP-GNN and comparison with existing methods
+
+In this section, we theoretically characterize the expressive power of KP-GNN and compare it with the original $K$ -hop message passing framework. The key insight is that, according to Equation (4), the message function at the $k$ -th hop additionally encodes $G _ { v , G } ^ { k , t }$ compared to normal $K$ -hop message passing. As we have already shown in the last section, -hop GNNs are bounded by 3-WL and thus cannot distinguish any non-isomorphic distance regular graphs, as well as Distance Encoding [22]. Let C k′ b e the $k ^ { \prime }$ -configuration of peripheral subgraph at $j$ -th hop of nodes in distance regular graph $G$ . Here we show that with the aid of peripheral subgraphs, KP-GNN is able to distinguish distance regular graphs:
+
+Proposition 2. For two non-isomorphic distance regular graphs $G ^ { ( 1 ) } = ( V ^ { ( 1 ) } , E ^ { ( 1 ) } )$ and $G ^ { ( 2 ) } =$ $( V ^ { ( 2 ) } , E ^ { ( 2 ) } )$ with the same diameter $d$ and intersection array $( b _ { 0 } , b _ { 1 } , . . . , b _ { d - 1 } ; c _ { 1 } , c _ { 2 } , . . . , c _ { d } )$ . Given $a$ er an $^ { l }$ - $d$ $K P$ th messafor some s defined in Equation (5), it can distinguish. $G ^ { ( 1 ) }$ $G ^ { ( 2 ) } i f C _ { j , G ^ { ( 1 ) } } ^ { k ^ { \prime } } \ne C _ { j , G ^ { ( 2 ) } } ^ { k ^ { \prime } }$ C k′j,G(2) 0 < j ≤ d
+
+![](images/a336ebea6c6a59a9a5ed56e6e6e2ef027008203798580fdef176ebc1b04f6672.jpg)  
+Figure 2: An example of two non-isomorphic distance regular graph with intersection array $( 6 , 3 ; 1 , 2 )$ .
+
+We include the proof in Appendix G. Here we leverage an example in Figure 2 to briefly show why KP-GNN is able to distinguish distance regular graphs. Figure 2 displays two distance regular graphs with an intersection array of $( 6 , 3 ; 1 , 2 )$ . The left one is Shrikhande graph and the right one is $4 \times 4$ Rook’s graph. Now, let’s look at the 1-hop peripheral subgraph of the green node. In the Shrikhande graph, there are 6 peripheral edges marked with red. Further, 6 edges constitute a circle. In the $4 \times 4$ Rook’s graph, there are still 6 peripheral edges. However, 6 edges constitute two circles with 3 edges in each circle, which is different from the Shrikhande graph. Then, any peripheral subgraph encoder that can distinguish these two graphs like node configuration enables the corresponding KP-GNN to distinguish the example. Proposition 2 shows that the KP-GNN is capable of distinguishing distance regular graphs, which further distinguishes KP-GNN from DE-1 [22] as it cannot distinguish any two connected distance regular graphs with the same intersection arrays according to Theorem 3.7 in [22]. However, it is currently unknown whether can KP-GNN with Equation 5 distinguish all distance regular graphs. We leave the detailed discussion in Appendix G.
+
+Moreover, both the subgraph-based GNNs like NGNN [26], GNN-AK [27], ESAN [28], and KPGNN leverage the information in the subgraph to enhance the power of message passing. However, KP-GNN is intrinsically different from them. Firstly, in KP-GNN, the message passing is performed on the whole graph instead of the subgraphs. This means that for each node, there is only one representation to be learned. Instead, for subgraph-based GNNs, the message passing is performed separately for each subgraph and each node could have multiple representations depending on which subgraph it is in. Secondly, in subgraph-based GNNs, they consider the subgraph as a whole without distinguishing nodes at different hops. Instead, KP-GNN takes one step further by dividing the subgraph into two parts. The first part is the hierarchy of neighbors at each hop. The second part is the connection structure between nodes in each hop. This gives us a better point of view to design a more powerful learning method. From the Corollary 7 in [23], we know that all subgraph-based GNNs with node selection as subgraph policy is bounded by 3-WL, which means they cannot distinguish any distance regular graph and KP-GNN is better at it.
+
+# 3.4 Time, space complexity, and limitation
+
+In this section, we discuss the time and space complexity of $K$ -hop message passing GNN and KPGNN. Suppose a graph has $n$ nodes and $m$ edges. Then, the $K$ -hop message passing and KP-GNN have both the space complexity of $O ( n )$ and the time complexity of $O ( \bar { n } ^ { 2 } )$ for the shortest path distance kernel. Note that the complexity of graph diffusion is no less than the shortest path distance kernel. We can see that KP-GNN only requires the same space complexity as vanilla GNNs and much less time complexity than the subgraph-based GNNs, which are at least $O ( n m )$ . However, $K$ -hop message passing including KP-GNN still have intrinsic limitation. We leave a detailed discussion on the complexity and limitation of KP-GNN in Appendix $_ \mathrm { H }$ .
+
+# 4 Related Work
+
+Expressive power of GNN. Analyzing the expressive power of GNNs is a crucial problem as it can serve as a guide on how to improve GNNs. Xu et al. [7] and Morris et al. [11] first proved that the power of 1-hop message passing is bounded by the 1-WL test. In other words, 1-hop message passing cannot distinguish any non-isomorphic graphs that the 1-WL test fails to. In recent years, many efforts have been put into increasing the expressive power of 1-hop messaging passing. The first line of research tries to mimic the higher-order WL tests, like 1-2-3 GNN [11], PPGN [24], ring-GNN [29]. However, they require exponentially increasing space and time complexity w.r.t. node number and cannot be generalized to large-scale graphs. The second line of research tries to enhance the rooted subtree of 1-WL with additional features. Some works [30, 31, 32] add one-hot or random features into nodes. Although they achieve good results in some settings, they deteriorate the generalization ability as such features produce different representations for nodes even with the same local graph structure. Some works like Distance Encoding [22], SEAL [33], labeling trick [34] and GLASS [35] introduce node labeling based on either distance or distinguishing target node set. On the other hand, GraphSNN [36] introduces a hierarchy of local isomorphism and proposes structural coefficients as additional features to identify such local isomorphism. However, the function designed to approximate the structural coefficient cannot fully achieve its theoretical power. The third line of research resorts to subgraph representation. Specifically, ID-GNN [37] extracts ego-netwok for each node and labels the root node with a different color. NGNN [26] encodes a rooted subgraph instead of a rooted subtree by subgraph pooling thus achieving superior expressive power on distinguishing regular graphs. GNN-AK [27] applies a similar idea as NGNN. The only difference lies in how to compute the node representation from the local subgraph. However, such methods need to run an inner GNN on every node of the graph thus introducing much more computation overhead. Meanwhile, the expressive power of subgraph GNNs are bounded by 3-WL [23].
+
+$K$ -hop message passing GNN. There are some existing works that instantiate the $K$ -hop message passing framework. For example, MixHop [12] performs message passing on each hop with graph diffusion kernel and concatenates the representation on each hop as the final representation. Khop [13] sequentially performs the message passing from hop K to hop 1 to compute the representation of the center node. However, it is not parallelizable due to its computational procedure. MAGNA [14] introduces an attention mechanism to $K$ -hop message passing. GPR-GNN [15] use graph diffusion kernel to perform graph convolution on $K$ -hop and aggregate them with learnable parameters. However, none of them give a formal definition of $K$ -hop message passing and theoretically analyze its representation power and limitations.
+
+# 5 Experiments
+
+In this section, we conduct extensive experiments to evaluate the performance of KP-GNN. Specifically, we 1) empirically verify the expressive power of KP-GNN on 3 simulation datasets and demonstrate the benefits of KP-GNN compared to normal $K$ -hop message passing GNNs; 2) demonstrate the effectiveness of KP-GNN on identifying various node properties, graph properties, and substructures with 3 simulation datasets; 3) show that the KP-GNN can achieve state-of-the-art performance on multiple real-world datasets; 4) analyze the running time of KP-GNN. The detail of each variant of KP-GNN is described in Appendix I and the detailed experimental setting is described in Appendix J. We implement the KP-GNN with PyTorch Geometric package [38]. Our code is available at https://github.com/JiaruiFeng/KP-GNN.
+
+Datasets: To evaluate the expressive power of KP-GNN, we choose: 1) EXP dataset [31], which contains 600 pairs of non-isomorphic graphs (1-WL failed). The goal is to map these graphs to two different classes. 2) SR25 dataset [39], which contains 15 non-isomorphic strongly regular graphs (3- WL failed) with each graph of 25 nodes. The dataset is translated to a 15-way classification problem with the goal of mapping each graph into different classes. 3) CSL dataset [40], which contains 150 4-regular graphs (1-WL failed) divided into 10 isomorphism classes. The goal of the task is to classify them into corresponding isomorphism classes. To demonstrate the capacity of KP-GNN on counting node/graph properties and substructures, we pick 1) Graph property regression (connectedness, diameter, radius) and node property regression (single source shortest path, eccentricity, Laplacian feature) task on random graph dataset [41]. 2) Graph substructure counting (triangle, tailed triangle, star, and 4-cycle) tasks on random graph dataset [42]. To evaluate the performance of KP-GNN on real-world datasets, we select 1) MUTAG [43], D&D [44], PROTEINS [44], PTC-MR [45], and IMDB-B [46] from TU database. 2) QM9 [47, 48] and ZINC [49] for molecular properties prediction. The detailed statistics of the datasets are described in Appendix L. Without further highlighting, all error bars in the result tables are the standard deviations of multiple runs.
+
+Table 1: Empirical evaluation of the expressive power.   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">K</td><td>EXP (ACC)</td><td>SR (ACC)</td><td></td><td>CSL (ACC)</td></tr><tr><td>SPD GD</td><td>SPD</td><td>GD</td><td>SPD GD</td></tr><tr><td rowspan="4">K-GIN</td><td>K=1</td><td>50 50</td><td>6.67</td><td>6.67</td><td>12</td></tr><tr><td>K=2</td><td>50 50</td><td>6.67</td><td>6.67</td><td>12 32 22.7</td></tr><tr><td>K=3</td><td>100 66.9</td><td>6.67</td><td>6.67</td><td>62 42</td></tr><tr><td>K=4</td><td>100 100</td><td>6.67</td><td>6.67</td><td>92.7 62.7</td></tr><tr><td rowspan="4">KP-GIN</td><td>K=1</td><td>50</td><td>50 100</td><td>100</td><td>22</td><td>22</td></tr><tr><td>K=2</td><td>100</td><td>100 100</td><td>100</td><td>52.7</td><td>52.7</td></tr><tr><td>K=3</td><td>100</td><td>100</td><td>100 100</td><td>90</td><td>90</td></tr><tr><td>K=4</td><td>100</td><td>100</td><td>100 100</td><td>100</td><td>100</td></tr></table>
+
+Empirical evaluation of the expressive power: For empirical evaluation of the expressive power, we conduct the ablation study on hop $K$ for both normal $K$ -hop GNNs and KP-GNN. For $K$ -hop GNNs, we implement K-GIN which uses GIN [7] as the base encoder. For KP-GNN, we implement KP-GIN. The results are shown in Table 1. Based on the results, we have the following conclusions: 1) $K$ -hop GNNs with both two kernels have expressive power higher than the 1-WL test as it shows the per
+
+fect performance on the EXP dataset and performance better than a random guess on the CSL dataset. 2) Increasing $K$ can improve the expressive for both two kernels. 3) $K$ -hop GNNs cannot distinguish any strong regular graphs in SR25 dataset, which is aligned with Theorem 2. 4) KP-GNN has much higher expressive power than normal $K$ -hop GNNs by showing better performance on every dataset given the same $K$ . Further, it achieves perfect results on the SR25 dataset even with $K = 1$ , which demonstrates its ability on distinguishing distance regular graphs.
+
+Table 2: Simulation dataset result. The top two are highlighted by First, Second.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Node Properties (log1o(MSE))</td><td colspan="3">Graph Properties (log1o(MSE))</td><td colspan="4">Counting Substructures (MAE)</td></tr><tr><td>SSSP</td><td>Ecc.</td><td>Lap.</td><td>Connect.</td><td>Diameter</td><td>Radius</td><td>Tri.</td><td>Tailed Tri.</td><td>Star</td><td>4-Cycle</td></tr><tr><td>GIN</td><td>-2.0000</td><td>-1.9000</td><td>-1.6000</td><td>-1.9239</td><td>-3.3079</td><td>-4.7584</td><td>0.3569</td><td>0.2373</td><td>0.0224</td><td>0.2185</td></tr><tr><td>PNA</td><td>-2.8900</td><td>-2.8900</td><td>-3.7700</td><td>-1.9395</td><td>3.4382</td><td>-4.9470</td><td>0.3532</td><td>0.2648</td><td>0.1278</td><td>0.2430</td></tr><tr><td>PPGN</td><td>-</td><td></td><td>-</td><td>-1.9804</td><td>-3.6147</td><td>-5.0878</td><td>0.0089</td><td>0.0096</td><td>0.0148</td><td>0.0090</td></tr><tr><td>GIN-AK+</td><td>-</td><td>-</td><td>-</td><td>-2.7513</td><td>-3.9687</td><td>-5.1846</td><td>0.0123</td><td>0.0112</td><td>0.0150</td><td>0.0126</td></tr><tr><td>K-GIN+</td><td>-2.7919</td><td>-2.5938</td><td>-4.6360</td><td>-2.1782</td><td>-3.9695</td><td>-5.3088</td><td>0.2593</td><td>0.1930</td><td>0.0165</td><td>0.2079</td></tr><tr><td>KP-GIN+</td><td>-2.7969</td><td>-2.6169</td><td>-4.7687</td><td>-4.4322</td><td>-3.9361</td><td>-5.3345</td><td>0.0060</td><td>0.0073</td><td>0.0151</td><td>0.0395</td></tr></table>
+
+Effectiveness on node/graph properties and substructure prediction: To evaluate the effectiveness of KP-GNN on node/graph properties and substructure prediction, we compare it with several existing models. For the baseline model, we use GIN [7], which has the same expressive power as the 1-WL test. For more powerful baselines, we use GIN-AK $^ +$ [27], PNA [41], and PPGN [24]. For normal $K$ -hop GNNs, we implement $\mathrm { K } { \mathrm { - G I N } } +$ , and for KP-GNN, we implement KP- $\mathrm { G I N + }$ . The results are shown in Table 2. Baseline results are taken from [27] and [41]. We can see ${ \mathrm { K P - G I N + } }$ achieve SOTA on a majority of tasks. Meanwhile, $\mathrm { K } { \mathrm { - G I N } } +$ also gets great performance on node/graph properties prediction. These results demonstrate the capability of KP-GNN to identify various properties and substructures. We leave the detailed results on counting substructures in Appendix K
+
+Table 3: TU dataset evaluation result.   
+
+<table><tr><td>Method</td><td>MUTAG</td><td>D&amp;D</td><td>PTC-MR</td><td>PROTEINS</td><td>IMDB-B</td></tr><tr><td>WL</td><td>90.4±5.7</td><td>79.4±0.3</td><td>59.9±4.3</td><td>75.0±3.1</td><td>73.8±3.9</td></tr><tr><td>GIN</td><td>89.4±5.6</td><td></td><td>64.6±7.0</td><td>75.9±2.8</td><td>75.1±5.1</td></tr><tr><td>DGCNN</td><td>85.8±1.7</td><td>79.3 ±0.9</td><td>58.6 ±2.5</td><td>75.5±0.9</td><td>70.0±0.9</td></tr><tr><td>GraphSNN</td><td>91.24±2.5</td><td>82.46±2.7</td><td>66.96±3.5</td><td>76.51±2.5</td><td>76.93±3.3</td></tr><tr><td>GIN-AK+</td><td>91.30±7.0</td><td>=</td><td>68.20±5.6</td><td>77.10±5.7</td><td>75.60±3.7</td></tr><tr><td>KP-GCN</td><td>91.7±6.0</td><td>79.0±4.7</td><td>67.1±6.3</td><td>75.8±3.5</td><td>75.9±3.8</td></tr><tr><td>KP-GraphSAGE</td><td>91.7±6.5</td><td>78.1±2.6</td><td>66.5±4.0</td><td>76.5±4.6</td><td>76.4±2.7</td></tr><tr><td>KP-GIN</td><td>92.2±6.5</td><td>79.4±3.8</td><td>66.8±6.8</td><td>75.8±4.6</td><td>76.6±4.2</td></tr><tr><td>GIN-AK+*</td><td>95.0±6.1</td><td>OOM</td><td>74.1±5.9</td><td>78.9±5.4</td><td>77.3±3.1</td></tr><tr><td>GraphSNN*</td><td>94.70±1.9</td><td>83.93±2.3</td><td>70.58±3.1</td><td>78.42±2.7</td><td>78.51±2.8</td></tr><tr><td>KP-GCN*</td><td>96.1±4.6</td><td>83.2±2.2</td><td>77.1±4.1</td><td>80.3±4.2</td><td>79.6±2.5</td></tr><tr><td>KP-GraphSAGE*</td><td>96.1±4.6</td><td>83.6±2.4</td><td>76.2±4.5</td><td>80.4±4.3</td><td>80.3±2.4</td></tr><tr><td>KP-GIN*</td><td>95.6±4.4</td><td>83.5±2.2</td><td>76.2±4.5</td><td>79.5±4.4</td><td>80.7±2.6</td></tr></table>
+
+Table 4: QM9 results. The top two are highlighted by First, Second.   
+
+<table><tr><td>Target</td><td>DTNN</td><td>MPNN</td><td>Deep LRP</td><td>PPGN</td><td>N-1-2-3-GNN</td><td>KP-GIN+</td><td>KP-GIN&#x27;</td></tr><tr><td>μ</td><td>0.244</td><td>0.358</td><td>0.364</td><td>0.231</td><td>0.433</td><td>0.367</td><td>0.358</td></tr><tr><td>α</td><td>0.95</td><td>0.89</td><td>0.298</td><td>0.382</td><td>0.265</td><td>0.242</td><td>0.233</td></tr><tr><td>εHOMO</td><td>0.00388</td><td>0.00541</td><td>0.00254</td><td>0.00276</td><td>0.00279</td><td>0.00247</td><td>0.00240</td></tr><tr><td>εLUMO</td><td>0.00512</td><td>0.00623</td><td>0.00277</td><td>0.00287</td><td>0.00276</td><td>0.00238</td><td>0.00236</td></tr><tr><td>△ε</td><td>0.0112</td><td>0.0066</td><td>0.00353</td><td>0.00406</td><td>0.00390</td><td>0.00345</td><td>0.00333</td></tr><tr><td>(R²)</td><td>17.0</td><td>28.5</td><td>19.3</td><td>16.7</td><td>20.1</td><td>16.49</td><td>16.51</td></tr><tr><td>ZPVE</td><td>0.00172</td><td>0.00216</td><td>0.00055</td><td>0.00064</td><td>0.00015</td><td>0.00018</td><td>0.00017</td></tr><tr><td>U</td><td>2.43</td><td>2.05</td><td>0.413</td><td>0.234</td><td>0.205</td><td>0.0728</td><td>0.0682</td></tr><tr><td>U</td><td>2.43</td><td>2.00</td><td>0.413</td><td>0.234</td><td>0.200</td><td>0.0553</td><td>0.0696</td></tr><tr><td>H</td><td>2.43</td><td>2.02</td><td>0.413</td><td>0.229</td><td>0.249</td><td>0.0575</td><td>0.0641</td></tr><tr><td>G</td><td>2.43</td><td>2.02</td><td>0.413</td><td>0.238</td><td>0.253</td><td>0.0526</td><td>0.0484</td></tr><tr><td>C</td><td>0.27</td><td>0.42</td><td>0.129</td><td>0.184</td><td>0.0811</td><td>0.0973</td><td>0.0869</td></tr></table>
+
+Evaluation on TU datasets: For baseline models, we select: 1) graph kernel-based method: WL subtree kernel [50]; 2) vanilla GNN methods: GIN [7] and DGCNN [6]; 3) advanced GNN methods: GraphSNN [36] and GIN- $\mathrm { \bf A K } +$ [27]. For the proposed KP-GNN, we implement GCN [1], GraphSAGE [3], and GIN [7] using the KP-GNN framework, denoted as KP-GCN, KP-GraphSAGE, and KP-GIN respectively. The results are shown in Table 3. For a more fair and comprehensive comparison, we report the results from two different evaluation settings. The first setting follows Xu et al. [7] and the second setting follows Wijesinghe and Wang [36]. We denote the second setting with ∗ in the table. We can see KP-GNN achieves SOTA performance on most of datasets under the second setting and still comparable performance to other baselines under the first setting.
+
+Table 5: ZINC result.   
+
+<table><tr><td>Method</td><td># param.</td><td>test MAE</td></tr><tr><td>MPNN</td><td>480805</td><td>0.145±0.007</td></tr><tr><td>PNA</td><td>387155</td><td>0.142±0.010</td></tr><tr><td>Graphormer</td><td>489321</td><td>0.122±0.006</td></tr><tr><td>GSN GIN-AK+</td><td>~500000</td><td>0.101±0.010 0.080±0.001</td></tr><tr><td>CIN</td><td>= 1</td><td>0.079±0.006</td></tr><tr><td>KP-GIN+</td><td>499099</td><td></td></tr><tr><td>KP-GIN&#x27;</td><td>488649</td><td>0.111±0.006 0.093±0.007</td></tr></table>
+
+Although KP-GNN does not achieve the best result, it is still comparable to other methods.
+
+Evaluation on molecular prediction tasks: For QM9 dataset, we report baseline results of DTNN and MPNN from [48]. We further select Deep LRP [42], PPGN [24], and Nested 1-2-3-GNN [26] as baseline models. For the ZINC dataset, we report results of MPNN [18] and PNA [41] from [17]. We further pick Graphormer [17], GSN [51], GIN-AK $^ +$ [27], and CIN [52]. For KP-GNN, we choose ${ \mathrm { K P - G I N } } +$ and ${ \bf K P - G I N } ^ { \prime }$ . The results of the QM9 dataset are shown in Table 4. We can see KPGNN achieves SOTA performance on most of the targets. The results of the ZINC dataset are shown in Table 5.
+
+Table 6: Running time (s/epoch).   
+
+<table><tr><td>Method</td><td>D&amp;D</td><td>ZINC</td><td>Graph property</td></tr><tr><td>GIN</td><td>1.10</td><td>3.59</td><td>1.02</td></tr><tr><td>K-GIN</td><td>3.94</td><td>6.44</td><td>1.67</td></tr><tr><td>KP-GIN</td><td>4.19</td><td>7.38</td><td>1.94</td></tr><tr><td>KP-GIN+</td><td>4.28</td><td>6.74</td><td>1.93</td></tr></table>
+
+the computational overhead is almost linear to $K$ . This is reasonable as practical graphs are sparse and the number of $K$ -hop neighbors is far less than $n$ when using a small $K$ .
+
+Running time comparison: In this section, we compare the running time of KP-GNN to 1-hop message passing GNN and $K$ -hop message passing GNN. We use GIN [7] as the base model. We also include the ${ \mathrm { K P - G I N + } }$ . All models use the same number of layers and hidden dimensions for a fair comparison. The results are shown in Table 6. We set $K = 4$ for all datasets. We can see
+
+# 6 Conclusion
+
+In this paper, we theoretically characterize the power of $K$ -hop message passing GNNs and propose the KP-GNN to improve the expressive power by leveraging the peripheral subgraph information at each hop. Theoretically, we prove that $K$ -hop GNNs can distinguish almost all regular graphs but are bounded by the 3-WL test. KP-GNN is able to distinguish many distance regular graphs. Empirically, KP-GNN achieves competitive results across all simulation and real-world datasets.
+
+# 7 Acknowledgement
+
+This work is partially supported by NSF grant CBE-2225809 and NSF China (No. 62276003).
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The implementation of KP-GNN can be found at https://github.com/JiaruiFeng/KP-GNN.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [No] The license can be found in their github.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] All datasets we used are all open-sourced.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] There is no personal information in the datasets.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/dev/nN3aVRQsxGd/nN3aVRQsxGd_content_list.json

@@ -0,0 +1,1503 @@
+[
+    {
+        "type": "text",
+        "text": "How Powerful are $K$ -hop Message Passing Graph Neural Networks ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jiarui Feng1,2 Yixin Chen1 Fuhai $\\mathbf { L i } ^ { 2 }$ Anindya Sarkar1 Muhan Zhang3,4 {feng.jiarui, fuhai.li, anindya}@wustl.edu, chen@cse.wustl.edu, muhan@pku.edu.cn 1Department of CSE, Washington University in St. Louis 2Institute for Informatics, Washington University School of Medicine 3Institute for Artificial Intelligence, Peking University 4Beijing Institute for General Artificial Intelligence ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The most popular design paradigm for Graph Neural Networks (GNNs) is 1-hop message passing—aggregating information from 1-hop neighbors repeatedly. However, the expressive power of 1-hop message passing is bounded by the WeisfeilerLehman (1-WL) test. Recently, researchers extended 1-hop message passing to $K$ -hop message passing by aggregating information from $K$ -hop neighbors of nodes simultaneously. However, there is no work on analyzing the expressive power of $K$ -hop message passing. In this work, we theoretically characterize the expressive power of $K$ -hop message passing. Specifically, we first formally differentiate two different kernels of $K$ -hop message passing which are often misused in previous works. We then characterize the expressive power of $K$ -hop message passing by showing that it is more powerful than 1-WL and can distinguish almost all regular graphs. Despite the higher expressive power, we show that $K$ -hop message passing still cannot distinguish some simple regular graphs and its expressive power is bounded by 3-WL. To further enhance its expressive power, we introduce a KP-GNN framework, which improves $K$ -hop message passing by leveraging the peripheral subgraph information in each hop. We show that KP-GNN can distinguish many distance regular graphs which could not be distinguished by previous distance encoding or 3-WL methods. Experimental results verify the expressive power and effectiveness of KP-GNN. KP-GNN achieves competitive results across all benchmark datasets. ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Currently, most existing graph neural networks (GNNs) follow the message passing framework, which iteratively aggregates information from the neighbors and updates the representations of nodes. It has shown superior performance on graph-related tasks [1, 2, 3, 4, 5, 6, 7] comparing to traditional graph embedding techniques [8, 9]. However, as the procedure of message passing is similar to the 1-dimensional Weisfeiler-Lehman (1-WL) test [10], the expressive power of message passing GNNs is also bounded by the 1-WL test [7, 11]. Namely, GNNs cannot distinguish two non-isomorphic graph structures if the 1-WL test fails. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In normal message passing GNNs, the node representation is updated by the direct neighbors of the node, which are called 1-hop neighbors. Recently, some works extend the notion of message passing into $K$ -hop message passing [12, 13, 14, 15, 16]. $K$ -hop message passing is a type of message passing where the node representation is updated by aggregating information from not only 1st hop but all the neighbors within $K$ hops of the node. However, there is no work on theoretically characterizing the expressive power of GNNs with $K$ -hop message passing, e.g., whether it can improve the 1-hop message passing or not and to what extent it can. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we theoretically characterize the expressive power of $K$ -hop message passing GNNs. Specifically, 1) we formally distinguish two different kernels of the $K$ -hop neighbors, which are often misused in previous works. The first kernel is based on whether the node can be reached within $k$ steps of the graph diffusion process, which is used in GPR-GNN [15] and MixHop [12]. The second one is based on the shortest path distance of $k$ , which is used in $\\mathrm { G I N E + }$ [16] and Graphormer [17]. Further, we show that different kernels of $K$ -hop neighbors will result in different expressive power of $K$ -hop message passing. 2) We show that $K$ -hop message passing is strictly more powerful than 1-hop message passing and can distinguish almost all regular graphs. 3) However, it still failed in distinguishing some simple regular graphs, no matter which kernel is used, and its expressive power is bounded by 3-WL. This motivates us to improve $K$ -hop message passing further. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Here, we introduce KP-GNN, a new GNN framework with $K$ -hop message passing, which significantly improves the expressive power of standard $K$ -hop message passing GNNs. In particular, during the aggregation of neighbors in each hop, KP-GNN not only aggregates neighboring nodes in that hop but also aggregates the peripheral subgraph (subgraph induced by the neighbors in that hop). This additional information helps the KP-GNN to learn more expressive local structural features around the node. We further show that KP-GNN can distinguish many distance regular graphs with a proper encoder for the peripheral subgraph. The proposed KP-GNN has several additional advantages. First, it can be applied to most existing $K$ -hop message-passing GNNs with only slight modification. Second, it only adds little computational complexity to standard $K$ -hop message passing. We demonstrate the effectiveness of the KP-GNN framework through extensive experiments on both simulation and real-world datasets. ",
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+    },
+    {
+        "type": "text",
+        "text": "2 $K$ -hop message passing and its expressive power ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "2.1 Notations ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Denote a graph as $G = ( V , E )$ , where $V = \\{ 1 , 2 , . . . , n \\}$ is the node set and $E \\subseteq V \\times V$ is the edge set. Meanwhile, denote $\\dot { A } \\in \\{ 0 , 1 \\} ^ { n \\times n }$ as the adjacency matrix of graph $G$ . Denote $x _ { v }$ as the feature vector of node $v$ and denote $e _ { u v }$ as the feature vector of the edge from $u$ to $v$ . Finally, we denote $Q _ { v , G } ^ { 1 }$ as the set of 1-hop neighbors of node $v$ in graph $G$ and $\\mathcal { N } _ { v , G } ^ { 1 ^ { - } } { = } Q _ { v , G } ^ { 1 } \\cup \\{ v \\}$ . Note that when we say $K$ -hop neighbors of node $v$ , we mean all the neighbors that have distance from node $v$ less than or equal to $K$ . In contrast, $k$ -th hop neighbors mean the neighbors with exactly distance $k$ from node $v$ . The definition of distance will be discussed in section 2.3. ",
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+    {
+        "type": "text",
+        "text": "2.2 1-hop message passing framework ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Currently, most existing GNNs are designed based on 1-hop message passing framework [18]. Denote $h _ { v } ^ { l }$ as the output representation of node $v$ at layer $l$ and $h _ { v } ^ { 0 } = x _ { v }$ . Briefly, given a graph $G$ and a 1-hop message passing GNN, at layer $l$ of the GNN, $h _ { v } ^ { l }$ is computed by $h _ { v } ^ { l - 1 }$ and $\\{ h _ { u } ^ { l - 1 } \\mid u \\in Q _ { v , G } ^ { 1 } \\}$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/305c0ee3f41fdb92d2e7e22ad51ff18f0cae65c73aa6024f10ddd7244a5dc726.jpg",
+        "text": "$$\nm _ { v } ^ { l } = \\mathrm { { \\bf M E S } } ^ { l } ( \\{ ( h _ { u } ^ { l - 1 } , e _ { u v } ) | u \\in Q _ { v , G } ^ { 1 } \\} ) , h _ { v } ^ { l } = \\mathrm { { \\bf U P D } } ^ { l } ( m _ { v } ^ { l } , h _ { v } ^ { l - 1 } ) ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $m _ { v } ^ { l }$ is the message to node $v$ at layer $l$ , $\\mathrm { M E S } ^ { l }$ and $\\mathrm { U P D } ^ { l }$ are message and update functions at layer $l$ respectively. After $L$ layers of message passing, $h _ { v } ^ { L }$ is used as the final representation of node $v$ . Such a representation can be used to conduct node-level tasks like node classification and node regression. To get the graph representation, a readout function is used: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5898975c9eda0a4e50611b10bc85bdc902a3af6c013579160cbed22c065ce94b.jpg",
+        "text": "$$\nh _ { G } = \\mathrm { R E A D O U T } ( \\{ h _ { v } ^ { L } | v \\in V \\} ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where READOUT is the readout function for computing the final graph representation. Then $h _ { G }$ can be used to conduct graph-level tasks like graph classification and graph regression. ",
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+    {
+        "type": "text",
+        "text": "2.3 $K$ -hop message passing framework ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The 1-hop message passing framework can be directly generalized to $K$ -hop message passing, as it shares the same message and update mechanism. The difference is that independent message and update functions can be employed for each hop. Meanwhile, a combination function is needed to combine the results from different hops into the final node representation at this layer. First, we differentiate two different kernels of $K$ -hop neighbors, which are interchanged and misused in previous research. ",
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+    },
+    {
+        "type": "text",
+        "text": "The first kernel of $K$ -hop neighbors is shortest path distance (spd) kernel. Namely, the $k$ -th hop neighbors of node $v$ in graph $G$ is the set of nodes with the shortest path distance of $k$ from $v$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Definition 1. For a node v in graph G, the K-hop neighbors N K,spdv,G of $v$ based on shortest path distance kernel is the set of nodes that have the shortest path distance from node $v$ less than or equal to $K$ . We further denote $Q _ { v , G } ^ { k , s p d }$ as the set of nodes in $G$ that are exactly the $k$ -th hop neighbors (with shortest path distance of exactly $k$ ) and $\\mathcal { N } _ { v , G } ^ { 0 , s p d } = Q _ { v , G } ^ { 0 , s p d } = \\{ v \\}$ is the node itself. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The second kernel of the $K$ -hop neighbors is based on graph diffusion $( g d )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 2. For a node v in graph $G$ , the $K$ -hop neighbors $\\mathcal { N } _ { v , G } ^ { K , g d }$ of v based on graph diffusion kernel is the sediffusion steps nodes that can diffuse iand the diffusion kernel ormation to node (adjacency matr $v$ within the number). We further denote dom walk as the set $K$ $A$ $Q _ { v , G } ^ { k , g d }$ of nodes in that are exactly the -th hop neighbors (nodes that can diffuse information to node with k diffusion steps) and N 0,gdv,G $\\mathcal { N } _ { v , G } ^ { 0 , g d } = Q _ { v , G } ^ { 0 , g \\bar { d } } = \\bar { \\{ v \\} }$ is the node itself. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Note that a node can be a $k$ -th hop neighbor of $v$ for multiple $k$ based on the graph diffusion kernel, but it can only appear in one hop for the shortest path distance kernel. We include more discussions of $K$ -hop kernels in Appendix A. Next, we define the $K$ -hop message passing framework as follows: ",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a41652654729b33de32ecc7a1984675370dc400f57ba2c723996e6ef1d4f30db.jpg",
+        "text": "$$\n\\begin{array} { r } { m _ { v } ^ { l , k } = \\mathrm { M E S } _ { k } ^ { l } ( \\{ ( h _ { u } ^ { l - 1 } , e _ { u v } ) | u \\in Q _ { v , G } ^ { k , t } ) \\} ) , h _ { v } ^ { l , k } = \\mathrm { U P D } _ { k } ^ { l } ( m _ { v } ^ { l , k } , h _ { v } ^ { l - 1 } ) , } \\\\ { h _ { v } ^ { l } = \\mathrm { C O M B I N E } ^ { l } ( \\{ \\{ h _ { v } ^ { l , k } | k = 1 , 2 , . . . , K \\} \\} ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            424,
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+            468
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $t \\in \\{ s p d , g d \\}$ , spd is the shortest path distance kernel and $_ { g d }$ is the graph diffusion kernel. Here, for each hop, we can apply unique MES and UPD functions. Note that for $k > 1$ , there may not exist the edge feature $e _ { u v }$ as nodes are not directly connected. But we leave it here since we can use other types of features to replace it like path encoding. We further discuss it in Appendix I. Compared to the 1-hop message passing framework described in Equation (1), the COMBINE function is introduced to combine the representations of node $v$ at different hops. It is easy to see that a $L$ layer 1-hop message passing GNNs is actually a $L$ layer $K$ -hop message passing GNNs with $K = 1$ . We include more discussions of $K$ -hop message passing GNNs in Appendix A. To aid further analysis, we also prove that $K$ -hop message passing can injectively encode the neighbor representations at different hops into $h _ { v } ^ { l }$ in Appendix B. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.4 Expressive power of $K$ -hop message passing framework ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we theoretically analyze the expressive power of $K$ -hop message passing. We assume there is no edge feature and all nodes in the graph have the same feature, which means that GNNs can only distinguish two nodes with the local structure of nodes. Note that including node features only increases the expressive power of GNNs as nodes/graphs are more easily to be discriminated. It has been proved that the expressive power of 1-hop message passing is bounded by the 1-WL test on discriminating non-isomorphic graphs [7, 11]. In this section, We show that the $K$ -hop message passing is strictly more powerful than the 1-WL test when $K > 1$ . Across the analysis, we utilize regular graphs as examples to illustrate our theorems since they cannot be distinguished using either 1-hop message passing or the 1-WL test. To begin the analysis, we first define proper $K$ -hop message passing GNNs. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 3. A proper $K$ -hop message passing GNN is a GNN model where the message, update, and combine functions are all injective given the input from a countable space. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "A proper $K$ -hop message passing GNN is easy to find due to the universal approximation theorem [19] of neural network and the Deep Set for set operation [20]. In the latter sections, by default, all mentioned $K$ -hop message passing GNNs are proper. Next, we introduce node configuration: ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 4. The node configuration of node v in graph $G$ within $K$ hops under t kernel is a list $A _ { v , G } ^ { K , t } = ( a _ { v , G } ^ { 1 , t } , a _ { v , G } ^ { 2 , t } , . . . , a _ { v , G } ^ { K , t } )$ , where $a _ { v , G } ^ { i , t } = | Q _ { v , G } ^ { i , t } |$ is the number of $i$ -th hop neighbors of node v. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/6717718d9a0f12e65c4bf76a241bcd84384ca02b31ebd9cd9710377c107a2563.jpg",
+        "image_caption": [
+            "Figure 1: Here are two pairs of non-isomorphic regular graphs. With 2-hop message passing, example 1 can be distinguished by the graph diffusion kernel, and example 2 can be distinguished by the shortest path distance kernel. However, both two examples become indistinguishable if we switch the kernel. Finally, both two examples can be distinguished by adding peripheral subgraph information. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "When we say two node configurations AK,t v1,G(1) and AK,t $A _ { v _ { 2 } , G ^ { ( 2 ) } } ^ { K , t }$ are equal, we mean that these two lists are component-wise equal to each other. Now, we state the first proposition: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. A proper $K$ -hop message passing GNN is strictly more powerful than 1-hop message passing GNNs when $K > 1$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To see why this is true, we first discuss how node configuration relates to the first layer of message passing. In the first layer of $K$ -hop message passing, each node aggregates neighbors from up to $K$ hops. As each node has the same node label, an injective message function can only know how many neighbors at each hop, which is exactly the node configuration. In other words, The first layer of $K$ -hop message passing is equivalent to inject node configuration to each node label. When $K = 1$ , the node configuration of $v _ { 1 }$ and $v _ { 2 }$ are $d _ { v _ { 1 } , G ^ { ( 1 ) } }$ and $d _ { v _ { 2 } , G ^ { ( 2 ) } }$ , where $d _ { v , G }$ is the node degree of $v$ . After $L$ layers, GNNs can only get the node degree information of each node within $L$ hops of node $v$ . Then, it is straightforward to see why these GNNs cannot distinguish any $n$ -sized $r$ -regular graph, as each node in the regular graph has the same degree. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Next, when $K > 1$ , the $K$ -hop message passing is at least equally powerful as 1-hop message passing since node configuration up to $K$ hop includes all the information the 1 hop has. To see why it is more powerful, we use two examples to illustrate it. The first example is shown in the left part of Figure 1. Suppose here we use graph diffusion kernel and we want to learn the representation of node $v _ { 1 }$ and node $v _ { 2 }$ in the two graphs. We know that the 1-hop GNNs produce the same representation for two nodes as they are both nodes in 6-sized 3-regular graphs. However, it is easy to see that $v _ { 1 }$ and $v _ { 2 }$ have different local structures and should have different representations. Instead, if we use the 2-hop message passing with the graph diffusion kernel, we can easily distinguish two nodes by checking the 2nd hop neighbors of the node, as node $v _ { 1 }$ has four 2nd hop neighbors but node $v _ { 2 }$ only has two 2nd hop neighbors. The second example is shown in the right part of Figure 1. Two graphs in the example are still regular graphs. Suppose here we use shortest path distance kernel, node $v _ { 1 }$ and $v _ { 2 }$ have different numbers of 2nd hop neighbors and thus will have different representations by performing 2-hop message passing. These two examples convincingly demonstrate that the $K$ -hop message passing with $K > 1$ can have better expressive power than $K = 1$ . To further study the expressive power of $K$ -hop message passing on regular graphs, we show the following result: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Consider all pairs of $n$ -sized $r$ -regular graphs, let $3 \\leq r < ( 2 l o g 2 n ) ^ { 1 / 2 }$ and $\\epsilon$ be a fixed constant. With at most $\\begin{array} { r } { K = \\lfloor ( \\frac { 1 } { 2 } + \\epsilon ) \\frac { \\log { 2 n } } { \\log { ( r - 1 ) } } \\rfloor } \\end{array}$ , there exists a $I$ layer $K$ -hop message passing GNN using the shortest path distance kernel that distinguishes almost all $1 - o ( n ^ { - 1 / 2 } )$ such pairs of graphs. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We include the proof and simulation results in Appendix C. Theorem 1 shows that even with 1 layer and a modest $K$ , $K$ -hop GNNs are powerful enough to distinguish almost all regular graphs. ",
+        "bbox": [
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+            882,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Finally, we characterize the existing $K$ -hop methods with the proposed $K$ -hop message passing framework. Specifically, we show that 1) the expressive power of $K$ layer GINE [16] is bounded by $K$ layer $K$ -hop message passing with the shortest path distance kernel. 2) The expressive power of the Graphormer [17] is equal to $K$ -hop GNNs with the shortest path distance kernel and infinity $K$ . 3) For spectral GNNs and existing $K$ -hop GNNs with the graph diffusion kernel like MixHop [12] and MAGNA [14], we find they actually use a weak version of $K$ -hop than the definition of us. Specifically, it is shown that the expressive power of spectral GNNs is also bounded by 1-WL test [21], which contradicts our result as graph diffusion can be viewed as a special case of spectral GNN. However, we show that our definition of $K$ -hop message passing with graph diffusion kernel actually injects a non-linear function on the spectral basis, thus achieving superior expressive power. We leave the detailed discussion in Appendix D. Further, Distance Encoding [22] also uses the shortest path distance information to augment the 1-hop message passing, which is similar to $K$ -hop GNNs with the shortest path distance kernel. However, we find the expressive power of the two frameworks differs from each other. We leave the detailed discussion in Appendix E. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.5 Limitation of $K$ -hop message passing framework ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Although we show that $K$ -hop GNNs with $K > 1$ are better at distinguishing non-isomorphic structures than 1-hop GNNs, there are still limitations. In this section, we discuss the limitation of $K$ -hop message passing. Specifically, we show that the choice of the kernel can affect the expressive power of $K$ -hop message passing. Furthermore, even with $K$ -hop message passing, we still cannot distinguish some simple non-isomorphic structures and the expressive power of $K$ -hop message passing is bounded by 3-WL. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Continue looking at the provided examples in Figure 1. In example 1, if we use the shortest path distance kernel instead of the graph diffusion kernel, two nodes have the same number of neighbors in the 2nd hop, which means that we cannot distinguish two nodes this time. Similarly, in example 2, two nodes have the same number of neighbors in both 1st and 2nd hops using graph diffusion kernel. These results highlight that the choice of the kernel can affect the expressive power of $K$ -hop message passing, and none of them can distinguish both two examples with 2-hop message passing. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Recently, Frasca et al. [23] show that any subgraph-based GNNs with node-based selection policy can be implemented by 3-IGN [24, 25] and thus their expressive power is bounded by 3-WL test. Here, we show that the $K$ -hop message passing GNNs can also be implemented by 3-IGN for both two kernels and thus: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. The expressive power of a proper $K$ -hop message passing GNN of any kernel is bounded by the 3-WL test. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We include the proof in Appendix F. Given all these observations, we may wonder if there is a way to further improve the expressive power of $K$ -hop message passing? ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 KP-GNN: improving the power of $K$ -hop message passing by peripheral subgraph ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            666,
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+            702
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we describe how to improve the expressive power of $K$ -hop message passing by adding additional information to the message passing framework. Specifically, by adding peripheral subgraph information, we can improve the expressive power of the $K$ -hop message passing by a large margin. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 Peripheral edge and peripheral subgraph ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            794,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "First, we define peripheral edge and peripheral subgraph. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 5. The peripheral edge $E ( Q _ { v , G } ^ { k , t } )$ is defined as the set of edges that connect nodes within set $Q _ { v , G } ^ { k , t }$ . We further denote $| E ( Q _ { v , G } ^ { k , t } ) |$ as the number of peripheral edge in $E ( Q _ { v , G } ^ { k , t } )$ . The peripheral subgraph Gk,tv,G $G _ { v , G } ^ { k , t } = ( Q _ { v , G } ^ { k , t } , E ( Q _ { v , G } ^ { k , t } ) )$ is defined as the subgraph induced by $Q _ { v , G } ^ { k , t }$ from the whole graph $G$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Briefly speaking, the peripheral edge $E ( Q _ { v , G } ^ { k , t } )$ record all the edges whose two ends are both from $Q _ { v , G } ^ { k , t }$ and the peripheral subgraph is a graph constituted by peripheral edges. It is easy to see that the peripheral subgraph $G _ { v , G } ^ { k , t }$ automatically contains all the information of peripheral edge $E ( Q _ { v , G } ^ { k , t } )$ . Next, we show that the power of $K$ -hop message passing can be improved by leveraging the information of peripheral edges and peripheral subgraphs. We again refer to the examples in Figure 1. Here we only consider the peripheral edge information. In example 1, we notice that at the 1st hop, there is an edge between node 3 and node 4 in the left graph. More specifically, $E ( Q _ { v _ { 1 } , G ^ { ( 1 ) } } ^ { 1 , t } ) \\stackrel { - } { = } \\{ ( 3 , 4 ) \\}$ . In contrast, we have $E ( Q _ { v _ { 2 } , G ^ { ( 2 ) } } ^ { 1 , t } ) = \\{ \\}$ in the right graph, which means there is no edge between the 1st hop neighbors of $v _ { 2 }$ . Therefore, we can successfully distinguish these two nodes by adding this information to the message passing. Similarly, in example 2, there is one edge between the 1st hop neighbors of node $v _ { 2 }$ , but no such edge exists for node $v _ { 1 }$ . By leveraging peripheral edge information, we can also distinguish the two nodes. The above examples demonstrate the effectiveness of the peripheral edge and peripheral subgraph information. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2 $K$ -hop peripheral-subgraph-enhanced graph neural network ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we propose $\\mathbf { K }$ -hop Peripheral-subgraph-enhanced Graph Neural Network (KP-GNN), which equips $K$ -hop message passing GNNs with peripheral subgraph information for more powerful GNN design. Recall the $K$ -hop message passing defined in Equation (3). The only difference between KP-GNN and original $K$ -hop GNNs is that we revise the message function as follows: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/15635446b9afcfc4cb80ceb9b745d310a817f09dcfd12361a672f257f57baf15.jpg",
+        "text": "$$\n\\begin{array} { r } { m _ { v } ^ { l , k } = \\mathbf { M E S } _ { k } ^ { l } ( \\{ ( h _ { u } ^ { l - 1 } , \\ e _ { u v } ) | u \\in Q _ { v , G } ^ { k , t } \\} , \\ G _ { v , G } ^ { k , t } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            402,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Briefly speaking, in the message step at the $k$ -th hop, we not only aggregate information of the neighbors but also the peripheral subgraph at that hop. The implementation of KP-GNN can be very flexible, as any graph encoding function can be used. To maximize the information the model can encode while keeping it simple, we implement the message function as: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9553c3d581bf6b39db2d10a049a422d4d393b91affd37c05257b4d366760d71e.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbf { M E S } _ { k } ^ { l } = \\mathbf { M E S } _ { k } ^ { l , n o r m a l } ( \\ P ( h _ { u } ^ { l - 1 } , e _ { u v } ) | u \\in Q _ { v , G } ^ { k , t } \\| ) + f ( G _ { v , G } ^ { k , t } ) , } \\\\ & { \\qquad f ( G _ { v , G } ^ { k , t } ) = \\mathbf { E M B } ( ( E ( Q _ { v , G } ^ { k , t } ) , C _ { k } ^ { k ^ { \\prime } } ) ) ~ , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            513,
+            705,
+            561
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where MESl,normal denotes the message function in the original GNN model, $C _ { k } ^ { k ^ { \\prime } }$ is the $k ^ { \\prime }$ configuration, which encode both node configuration and the number of the peripheral edge of all nodes in $G _ { v , G } ^ { k , t }$ up to $k ^ { \\prime }$ hops. It can be regarded as running another 1 layer KP-GNN and readout function on each peripheral subgraph. EMB is a learnable embedding function. With this implementation, any base GNN model can be incorporated into and be enhanced by the KP-GNN framework by replacing $\\mathbf { M E S } _ { k } ^ { l , n o r m a l }$ and $\\mathrm { U P D } _ { k } ^ { l }$ with the corresponding functions for each hop $k$ . We leave the detailed implementation in Appendix I. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3 The expressive power of KP-GNN and comparison with existing methods ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            700,
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+            717
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we theoretically characterize the expressive power of KP-GNN and compare it with the original $K$ -hop message passing framework. The key insight is that, according to Equation (4), the message function at the $k$ -th hop additionally encodes $G _ { v , G } ^ { k , t }$ compared to normal $K$ -hop message passing. As we have already shown in the last section, -hop GNNs are bounded by 3-WL and thus cannot distinguish any non-isomorphic distance regular graphs, as well as Distance Encoding [22]. Let C k′ b e the $k ^ { \\prime }$ -configuration of peripheral subgraph at $j$ -th hop of nodes in distance regular graph $G$ . Here we show that with the aid of peripheral subgraphs, KP-GNN is able to distinguish distance regular graphs: ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2. For two non-isomorphic distance regular graphs $G ^ { ( 1 ) } = ( V ^ { ( 1 ) } , E ^ { ( 1 ) } )$ and $G ^ { ( 2 ) } =$ $( V ^ { ( 2 ) } , E ^ { ( 2 ) } )$ with the same diameter $d$ and intersection array $( b _ { 0 } , b _ { 1 } , . . . , b _ { d - 1 } ; c _ { 1 } , c _ { 2 } , . . . , c _ { d } )$ . Given $a$ er an $^ { l }$ - $d$ $K P$ th messafor some s defined in Equation (5), it can distinguish. $G ^ { ( 1 ) }$ $G ^ { ( 2 ) } i f C _ { j , G ^ { ( 1 ) } } ^ { k ^ { \\prime } } \\ne C _ { j , G ^ { ( 2 ) } } ^ { k ^ { \\prime } }$ C k′j,G(2) 0 < j ≤ d ",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/a336ebea6c6a59a9a5ed56e6e6e2ef027008203798580fdef176ebc1b04f6672.jpg",
+        "image_caption": [
+            "Figure 2: An example of two non-isomorphic distance regular graph with intersection array $( 6 , 3 ; 1 , 2 )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We include the proof in Appendix G. Here we leverage an example in Figure 2 to briefly show why KP-GNN is able to distinguish distance regular graphs. Figure 2 displays two distance regular graphs with an intersection array of $( 6 , 3 ; 1 , 2 )$ . The left one is Shrikhande graph and the right one is $4 \\times 4$ Rook’s graph. Now, let’s look at the 1-hop peripheral subgraph of the green node. In the Shrikhande graph, there are 6 peripheral edges marked with red. Further, 6 edges constitute a circle. In the $4 \\times 4$ Rook’s graph, there are still 6 peripheral edges. However, 6 edges constitute two circles with 3 edges in each circle, which is different from the Shrikhande graph. Then, any peripheral subgraph encoder that can distinguish these two graphs like node configuration enables the corresponding KP-GNN to distinguish the example. Proposition 2 shows that the KP-GNN is capable of distinguishing distance regular graphs, which further distinguishes KP-GNN from DE-1 [22] as it cannot distinguish any two connected distance regular graphs with the same intersection arrays according to Theorem 3.7 in [22]. However, it is currently unknown whether can KP-GNN with Equation 5 distinguish all distance regular graphs. We leave the detailed discussion in Appendix G. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Moreover, both the subgraph-based GNNs like NGNN [26], GNN-AK [27], ESAN [28], and KPGNN leverage the information in the subgraph to enhance the power of message passing. However, KP-GNN is intrinsically different from them. Firstly, in KP-GNN, the message passing is performed on the whole graph instead of the subgraphs. This means that for each node, there is only one representation to be learned. Instead, for subgraph-based GNNs, the message passing is performed separately for each subgraph and each node could have multiple representations depending on which subgraph it is in. Secondly, in subgraph-based GNNs, they consider the subgraph as a whole without distinguishing nodes at different hops. Instead, KP-GNN takes one step further by dividing the subgraph into two parts. The first part is the hierarchy of neighbors at each hop. The second part is the connection structure between nodes in each hop. This gives us a better point of view to design a more powerful learning method. From the Corollary 7 in [23], we know that all subgraph-based GNNs with node selection as subgraph policy is bounded by 3-WL, which means they cannot distinguish any distance regular graph and KP-GNN is better at it. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.4 Time, space complexity, and limitation ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we discuss the time and space complexity of $K$ -hop message passing GNN and KPGNN. Suppose a graph has $n$ nodes and $m$ edges. Then, the $K$ -hop message passing and KP-GNN have both the space complexity of $O ( n )$ and the time complexity of $O ( \\bar { n } ^ { 2 } )$ for the shortest path distance kernel. Note that the complexity of graph diffusion is no less than the shortest path distance kernel. We can see that KP-GNN only requires the same space complexity as vanilla GNNs and much less time complexity than the subgraph-based GNNs, which are at least $O ( n m )$ . However, $K$ -hop message passing including KP-GNN still have intrinsic limitation. We leave a detailed discussion on the complexity and limitation of KP-GNN in Appendix $_ \\mathrm { H }$ . ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Expressive power of GNN. Analyzing the expressive power of GNNs is a crucial problem as it can serve as a guide on how to improve GNNs. Xu et al. [7] and Morris et al. [11] first proved that the power of 1-hop message passing is bounded by the 1-WL test. In other words, 1-hop message passing cannot distinguish any non-isomorphic graphs that the 1-WL test fails to. In recent years, many efforts have been put into increasing the expressive power of 1-hop messaging passing. The first line of research tries to mimic the higher-order WL tests, like 1-2-3 GNN [11], PPGN [24], ring-GNN [29]. However, they require exponentially increasing space and time complexity w.r.t. node number and cannot be generalized to large-scale graphs. The second line of research tries to enhance the rooted subtree of 1-WL with additional features. Some works [30, 31, 32] add one-hot or random features into nodes. Although they achieve good results in some settings, they deteriorate the generalization ability as such features produce different representations for nodes even with the same local graph structure. Some works like Distance Encoding [22], SEAL [33], labeling trick [34] and GLASS [35] introduce node labeling based on either distance or distinguishing target node set. On the other hand, GraphSNN [36] introduces a hierarchy of local isomorphism and proposes structural coefficients as additional features to identify such local isomorphism. However, the function designed to approximate the structural coefficient cannot fully achieve its theoretical power. The third line of research resorts to subgraph representation. Specifically, ID-GNN [37] extracts ego-netwok for each node and labels the root node with a different color. NGNN [26] encodes a rooted subgraph instead of a rooted subtree by subgraph pooling thus achieving superior expressive power on distinguishing regular graphs. GNN-AK [27] applies a similar idea as NGNN. The only difference lies in how to compute the node representation from the local subgraph. However, such methods need to run an inner GNN on every node of the graph thus introducing much more computation overhead. Meanwhile, the expressive power of subgraph GNNs are bounded by 3-WL [23]. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "$K$ -hop message passing GNN. There are some existing works that instantiate the $K$ -hop message passing framework. For example, MixHop [12] performs message passing on each hop with graph diffusion kernel and concatenates the representation on each hop as the final representation. Khop [13] sequentially performs the message passing from hop K to hop 1 to compute the representation of the center node. However, it is not parallelizable due to its computational procedure. MAGNA [14] introduces an attention mechanism to $K$ -hop message passing. GPR-GNN [15] use graph diffusion kernel to perform graph convolution on $K$ -hop and aggregate them with learnable parameters. However, none of them give a formal definition of $K$ -hop message passing and theoretically analyze its representation power and limitations. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we conduct extensive experiments to evaluate the performance of KP-GNN. Specifically, we 1) empirically verify the expressive power of KP-GNN on 3 simulation datasets and demonstrate the benefits of KP-GNN compared to normal $K$ -hop message passing GNNs; 2) demonstrate the effectiveness of KP-GNN on identifying various node properties, graph properties, and substructures with 3 simulation datasets; 3) show that the KP-GNN can achieve state-of-the-art performance on multiple real-world datasets; 4) analyze the running time of KP-GNN. The detail of each variant of KP-GNN is described in Appendix I and the detailed experimental setting is described in Appendix J. We implement the KP-GNN with PyTorch Geometric package [38]. Our code is available at https://github.com/JiaruiFeng/KP-GNN. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Datasets: To evaluate the expressive power of KP-GNN, we choose: 1) EXP dataset [31], which contains 600 pairs of non-isomorphic graphs (1-WL failed). The goal is to map these graphs to two different classes. 2) SR25 dataset [39], which contains 15 non-isomorphic strongly regular graphs (3- WL failed) with each graph of 25 nodes. The dataset is translated to a 15-way classification problem with the goal of mapping each graph into different classes. 3) CSL dataset [40], which contains 150 4-regular graphs (1-WL failed) divided into 10 isomorphism classes. The goal of the task is to classify them into corresponding isomorphism classes. To demonstrate the capacity of KP-GNN on counting node/graph properties and substructures, we pick 1) Graph property regression (connectedness, diameter, radius) and node property regression (single source shortest path, eccentricity, Laplacian feature) task on random graph dataset [41]. 2) Graph substructure counting (triangle, tailed triangle, star, and 4-cycle) tasks on random graph dataset [42]. To evaluate the performance of KP-GNN on real-world datasets, we select 1) MUTAG [43], D&D [44], PROTEINS [44], PTC-MR [45], and IMDB-B [46] from TU database. 2) QM9 [47, 48] and ZINC [49] for molecular properties prediction. The detailed statistics of the datasets are described in Appendix L. Without further highlighting, all error bars in the result tables are the standard deviations of multiple runs. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c5c197f604498b7b6279bb791e4f6a515fa6dcb7fe0d5acc961163d7bd52dd26.jpg",
+        "table_caption": [
+            "Table 1: Empirical evaluation of the expressive power. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">K</td><td>EXP (ACC)</td><td>SR (ACC)</td><td></td><td>CSL (ACC)</td></tr><tr><td>SPD GD</td><td>SPD</td><td>GD</td><td>SPD GD</td></tr><tr><td rowspan=\"4\">K-GIN</td><td>K=1</td><td>50 50</td><td>6.67</td><td>6.67</td><td>12</td></tr><tr><td>K=2</td><td>50 50</td><td>6.67</td><td>6.67</td><td>12 32 22.7</td></tr><tr><td>K=3</td><td>100 66.9</td><td>6.67</td><td>6.67</td><td>62 42</td></tr><tr><td>K=4</td><td>100 100</td><td>6.67</td><td>6.67</td><td>92.7 62.7</td></tr><tr><td rowspan=\"4\">KP-GIN</td><td>K=1</td><td>50</td><td>50 100</td><td>100</td><td>22</td><td>22</td></tr><tr><td>K=2</td><td>100</td><td>100 100</td><td>100</td><td>52.7</td><td>52.7</td></tr><tr><td>K=3</td><td>100</td><td>100</td><td>100 100</td><td>90</td><td>90</td></tr><tr><td>K=4</td><td>100</td><td>100</td><td>100 100</td><td>100</td><td>100</td></tr></table>",
+        "bbox": [
+            176,
+            175,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Empirical evaluation of the expressive power: For empirical evaluation of the expressive power, we conduct the ablation study on hop $K$ for both normal $K$ -hop GNNs and KP-GNN. For $K$ -hop GNNs, we implement K-GIN which uses GIN [7] as the base encoder. For KP-GNN, we implement KP-GIN. The results are shown in Table 1. Based on the results, we have the following conclusions: 1) $K$ -hop GNNs with both two kernels have expressive power higher than the 1-WL test as it shows the per",
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+    {
+        "type": "text",
+        "text": "fect performance on the EXP dataset and performance better than a random guess on the CSL dataset. 2) Increasing $K$ can improve the expressive for both two kernels. 3) $K$ -hop GNNs cannot distinguish any strong regular graphs in SR25 dataset, which is aligned with Theorem 2. 4) KP-GNN has much higher expressive power than normal $K$ -hop GNNs by showing better performance on every dataset given the same $K$ . Further, it achieves perfect results on the SR25 dataset even with $K = 1$ , which demonstrates its ability on distinguishing distance regular graphs. ",
+        "bbox": [
+            173,
+            319,
+            825,
+            402
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/1a643e9cf5841554acebc8a4a07a4a1e8b9b2209d0fa83e58c9c5f27a5a4bb9f.jpg",
+        "table_caption": [
+            "Table 2: Simulation dataset result. The top two are highlighted by First, Second. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">Node Properties (log1o(MSE))</td><td colspan=\"3\">Graph Properties (log1o(MSE))</td><td colspan=\"4\">Counting Substructures (MAE)</td></tr><tr><td>SSSP</td><td>Ecc.</td><td>Lap.</td><td>Connect.</td><td>Diameter</td><td>Radius</td><td>Tri.</td><td>Tailed Tri.</td><td>Star</td><td>4-Cycle</td></tr><tr><td>GIN</td><td>-2.0000</td><td>-1.9000</td><td>-1.6000</td><td>-1.9239</td><td>-3.3079</td><td>-4.7584</td><td>0.3569</td><td>0.2373</td><td>0.0224</td><td>0.2185</td></tr><tr><td>PNA</td><td>-2.8900</td><td>-2.8900</td><td>-3.7700</td><td>-1.9395</td><td>3.4382</td><td>-4.9470</td><td>0.3532</td><td>0.2648</td><td>0.1278</td><td>0.2430</td></tr><tr><td>PPGN</td><td>-</td><td></td><td>-</td><td>-1.9804</td><td>-3.6147</td><td>-5.0878</td><td>0.0089</td><td>0.0096</td><td>0.0148</td><td>0.0090</td></tr><tr><td>GIN-AK+</td><td>-</td><td>-</td><td>-</td><td>-2.7513</td><td>-3.9687</td><td>-5.1846</td><td>0.0123</td><td>0.0112</td><td>0.0150</td><td>0.0126</td></tr><tr><td>K-GIN+</td><td>-2.7919</td><td>-2.5938</td><td>-4.6360</td><td>-2.1782</td><td>-3.9695</td><td>-5.3088</td><td>0.2593</td><td>0.1930</td><td>0.0165</td><td>0.2079</td></tr><tr><td>KP-GIN+</td><td>-2.7969</td><td>-2.6169</td><td>-4.7687</td><td>-4.4322</td><td>-3.9361</td><td>-5.3345</td><td>0.0060</td><td>0.0073</td><td>0.0151</td><td>0.0395</td></tr></table>",
+        "bbox": [
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+            424,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Effectiveness on node/graph properties and substructure prediction: To evaluate the effectiveness of KP-GNN on node/graph properties and substructure prediction, we compare it with several existing models. For the baseline model, we use GIN [7], which has the same expressive power as the 1-WL test. For more powerful baselines, we use GIN-AK $^ +$ [27], PNA [41], and PPGN [24]. For normal $K$ -hop GNNs, we implement $\\mathrm { K } { \\mathrm { - G I N } } +$ , and for KP-GNN, we implement KP- $\\mathrm { G I N + }$ . The results are shown in Table 2. Baseline results are taken from [27] and [41]. We can see ${ \\mathrm { K P - G I N + } }$ achieve SOTA on a majority of tasks. Meanwhile, $\\mathrm { K } { \\mathrm { - G I N } } +$ also gets great performance on node/graph properties prediction. These results demonstrate the capability of KP-GNN to identify various properties and substructures. We leave the detailed results on counting substructures in Appendix K ",
+        "bbox": [
+            173,
+            542,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/45eddb1792f89b8a133a3ef6b662f1514f9f081869bad15bc8a6eda57df33ce9.jpg",
+        "table_caption": [
+            "Table 3: TU dataset evaluation result. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>MUTAG</td><td>D&amp;D</td><td>PTC-MR</td><td>PROTEINS</td><td>IMDB-B</td></tr><tr><td>WL</td><td>90.4±5.7</td><td>79.4±0.3</td><td>59.9±4.3</td><td>75.0±3.1</td><td>73.8±3.9</td></tr><tr><td>GIN</td><td>89.4±5.6</td><td></td><td>64.6±7.0</td><td>75.9±2.8</td><td>75.1±5.1</td></tr><tr><td>DGCNN</td><td>85.8±1.7</td><td>79.3 ±0.9</td><td>58.6 ±2.5</td><td>75.5±0.9</td><td>70.0±0.9</td></tr><tr><td>GraphSNN</td><td>91.24±2.5</td><td>82.46±2.7</td><td>66.96±3.5</td><td>76.51±2.5</td><td>76.93±3.3</td></tr><tr><td>GIN-AK+</td><td>91.30±7.0</td><td>=</td><td>68.20±5.6</td><td>77.10±5.7</td><td>75.60±3.7</td></tr><tr><td>KP-GCN</td><td>91.7±6.0</td><td>79.0±4.7</td><td>67.1±6.3</td><td>75.8±3.5</td><td>75.9±3.8</td></tr><tr><td>KP-GraphSAGE</td><td>91.7±6.5</td><td>78.1±2.6</td><td>66.5±4.0</td><td>76.5±4.6</td><td>76.4±2.7</td></tr><tr><td>KP-GIN</td><td>92.2±6.5</td><td>79.4±3.8</td><td>66.8±6.8</td><td>75.8±4.6</td><td>76.6±4.2</td></tr><tr><td>GIN-AK+*</td><td>95.0±6.1</td><td>OOM</td><td>74.1±5.9</td><td>78.9±5.4</td><td>77.3±3.1</td></tr><tr><td>GraphSNN*</td><td>94.70±1.9</td><td>83.93±2.3</td><td>70.58±3.1</td><td>78.42±2.7</td><td>78.51±2.8</td></tr><tr><td>KP-GCN*</td><td>96.1±4.6</td><td>83.2±2.2</td><td>77.1±4.1</td><td>80.3±4.2</td><td>79.6±2.5</td></tr><tr><td>KP-GraphSAGE*</td><td>96.1±4.6</td><td>83.6±2.4</td><td>76.2±4.5</td><td>80.4±4.3</td><td>80.3±2.4</td></tr><tr><td>KP-GIN*</td><td>95.6±4.4</td><td>83.5±2.2</td><td>76.2±4.5</td><td>79.5±4.4</td><td>80.7±2.6</td></tr></table>",
+        "bbox": [
+            209,
+            694,
+            761,
+            915
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/f73a7ff78f4d5026f9c3c6fe389493d38f3b66365f5d321b2bfd0a1b499d5eda.jpg",
+        "table_caption": [
+            "Table 4: QM9 results. The top two are highlighted by First, Second. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Target</td><td>DTNN</td><td>MPNN</td><td>Deep LRP</td><td>PPGN</td><td>N-1-2-3-GNN</td><td>KP-GIN+</td><td>KP-GIN&#x27;</td></tr><tr><td>μ</td><td>0.244</td><td>0.358</td><td>0.364</td><td>0.231</td><td>0.433</td><td>0.367</td><td>0.358</td></tr><tr><td>α</td><td>0.95</td><td>0.89</td><td>0.298</td><td>0.382</td><td>0.265</td><td>0.242</td><td>0.233</td></tr><tr><td>εHOMO</td><td>0.00388</td><td>0.00541</td><td>0.00254</td><td>0.00276</td><td>0.00279</td><td>0.00247</td><td>0.00240</td></tr><tr><td>εLUMO</td><td>0.00512</td><td>0.00623</td><td>0.00277</td><td>0.00287</td><td>0.00276</td><td>0.00238</td><td>0.00236</td></tr><tr><td>△ε</td><td>0.0112</td><td>0.0066</td><td>0.00353</td><td>0.00406</td><td>0.00390</td><td>0.00345</td><td>0.00333</td></tr><tr><td>(R²)</td><td>17.0</td><td>28.5</td><td>19.3</td><td>16.7</td><td>20.1</td><td>16.49</td><td>16.51</td></tr><tr><td>ZPVE</td><td>0.00172</td><td>0.00216</td><td>0.00055</td><td>0.00064</td><td>0.00015</td><td>0.00018</td><td>0.00017</td></tr><tr><td>U</td><td>2.43</td><td>2.05</td><td>0.413</td><td>0.234</td><td>0.205</td><td>0.0728</td><td>0.0682</td></tr><tr><td>U</td><td>2.43</td><td>2.00</td><td>0.413</td><td>0.234</td><td>0.200</td><td>0.0553</td><td>0.0696</td></tr><tr><td>H</td><td>2.43</td><td>2.02</td><td>0.413</td><td>0.229</td><td>0.249</td><td>0.0575</td><td>0.0641</td></tr><tr><td>G</td><td>2.43</td><td>2.02</td><td>0.413</td><td>0.238</td><td>0.253</td><td>0.0526</td><td>0.0484</td></tr><tr><td>C</td><td>0.27</td><td>0.42</td><td>0.129</td><td>0.184</td><td>0.0811</td><td>0.0973</td><td>0.0869</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Evaluation on TU datasets: For baseline models, we select: 1) graph kernel-based method: WL subtree kernel [50]; 2) vanilla GNN methods: GIN [7] and DGCNN [6]; 3) advanced GNN methods: GraphSNN [36] and GIN- $\\mathrm { \\bf A K } +$ [27]. For the proposed KP-GNN, we implement GCN [1], GraphSAGE [3], and GIN [7] using the KP-GNN framework, denoted as KP-GCN, KP-GraphSAGE, and KP-GIN respectively. The results are shown in Table 3. For a more fair and comprehensive comparison, we report the results from two different evaluation settings. The first setting follows Xu et al. [7] and the second setting follows Wijesinghe and Wang [36]. We denote the second setting with ∗ in the table. We can see KP-GNN achieves SOTA performance on most of datasets under the second setting and still comparable performance to other baselines under the first setting. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/8a8da0ef759bd005e37de99b7517d17daec720307d1327faafce3e6bcf9f01d4.jpg",
+        "table_caption": [
+            "Table 5: ZINC result. "
+        ],
+        "table_footnote": [
+            "Although KP-GNN does not achieve the best result, it is still comparable to other methods. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td># param.</td><td>test MAE</td></tr><tr><td>MPNN</td><td>480805</td><td>0.145±0.007</td></tr><tr><td>PNA</td><td>387155</td><td>0.142±0.010</td></tr><tr><td>Graphormer</td><td>489321</td><td>0.122±0.006</td></tr><tr><td>GSN GIN-AK+</td><td>~500000</td><td>0.101±0.010 0.080±0.001</td></tr><tr><td>CIN</td><td>= 1</td><td>0.079±0.006</td></tr><tr><td>KP-GIN+</td><td>499099</td><td></td></tr><tr><td>KP-GIN&#x27;</td><td>488649</td><td>0.111±0.006 0.093±0.007</td></tr></table>",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Evaluation on molecular prediction tasks: For QM9 dataset, we report baseline results of DTNN and MPNN from [48]. We further select Deep LRP [42], PPGN [24], and Nested 1-2-3-GNN [26] as baseline models. For the ZINC dataset, we report results of MPNN [18] and PNA [41] from [17]. We further pick Graphormer [17], GSN [51], GIN-AK $^ +$ [27], and CIN [52]. For KP-GNN, we choose ${ \\mathrm { K P - G I N } } +$ and ${ \\bf K P - G I N } ^ { \\prime }$ . The results of the QM9 dataset are shown in Table 4. We can see KPGNN achieves SOTA performance on most of the targets. The results of the ZINC dataset are shown in Table 5. ",
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+        "type": "table",
+        "img_path": "images/f293a9c2ff1c52ccb234265388bff3353706b20b5f3b57becfd8607c19e9e3d2.jpg",
+        "table_caption": [
+            "Table 6: Running time (s/epoch). "
+        ],
+        "table_footnote": [
+            "the computational overhead is almost linear to $K$ . This is reasonable as practical graphs are sparse and the number of $K$ -hop neighbors is far less than $n$ when using a small $K$ . "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>D&amp;D</td><td>ZINC</td><td>Graph property</td></tr><tr><td>GIN</td><td>1.10</td><td>3.59</td><td>1.02</td></tr><tr><td>K-GIN</td><td>3.94</td><td>6.44</td><td>1.67</td></tr><tr><td>KP-GIN</td><td>4.19</td><td>7.38</td><td>1.94</td></tr><tr><td>KP-GIN+</td><td>4.28</td><td>6.74</td><td>1.93</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Running time comparison: In this section, we compare the running time of KP-GNN to 1-hop message passing GNN and $K$ -hop message passing GNN. We use GIN [7] as the base model. We also include the ${ \\mathrm { K P - G I N + } }$ . All models use the same number of layers and hidden dimensions for a fair comparison. The results are shown in Table 6. We set $K = 4$ for all datasets. We can see ",
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+        "type": "text",
+        "text": "6 Conclusion ",
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+    {
+        "type": "text",
+        "text": "In this paper, we theoretically characterize the power of $K$ -hop message passing GNNs and propose the KP-GNN to improve the expressive power by leveraging the peripheral subgraph information at each hop. Theoretically, we prove that $K$ -hop GNNs can distinguish almost all regular graphs but are bounded by the 3-WL test. KP-GNN is able to distinguish many distance regular graphs. Empirically, KP-GNN achieves competitive results across all simulation and real-world datasets. ",
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+        "text": "7 Acknowledgement ",
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+        "text": "This work is partially supported by NSF grant CBE-2225809 and NSF China (No. 62276003). ",
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+        "text": "References   \n[1] Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. In International Conference on Learning Representations, 2017.   \n[2] David K Duvenaud, Dougal Maclaurin, Jorge Iparraguirre, Rafael Bombarell, Timothy Hirzel, Alán Aspuru-Guzik, and Ryan P Adams. Convolutional networks on graphs for learning molecular fingerprints. In Advances in neural information processing systems, pages 2224– 2232, 2015.   \n[3] Will Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs. In Advances in Neural Information Processing Systems, pages 1025–1035, 2017.   \n[4] Petar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua ´ Bengio. Graph attention networks. In International Conference on Learning Representations, 2018. URL https://openreview.net/forum?id=rJXMpikCZ.   \n[5] Yujia Li, Daniel Tarlow, Marc Brockschmidt, and Richard Zemel. Gated graph sequence neural networks. arXiv preprint arXiv:1511.05493, 2015.   \n[6] Muhan Zhang, Zhicheng Cui, Marion Neumann, and Yixin Chen. An end-to-end deep learning architecture for graph classification. In AAAI, pages 4438–4445, 2018.   \n[7] Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? In International Conference on Learning Representations, 2019. URL https: //openreview.net/forum?id $\\equiv$ ryGs6iA5Km.   \n[8] Aditya Grover and Jure Leskovec. node2vec: Scalable feature learning for networks. In Proceedings of the 22nd ACM SIGKDD international conference on Knowledge discovery and data mining, pages 855–864. ACM, 2016.   \n[9] Bryan Perozzi, Rami Al-Rfou, and Steven Skiena. Deepwalk: Online learning of social representations. In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 701–710. ACM, 2014.   \n[10] Boris Weisfeiler and AA Lehman. A reduction of a graph to a canonical form and an algebra arising during this reduction. Nauchno-Technicheskaya Informatsia, 2(9):12–16, 1968.   \n[11] Christopher Morris, Martin Ritzert, Matthias Fey, William L Hamilton, Jan Eric Lenssen, Gaurav Rattan, and Martin Grohe. Weisfeiler and leman go neural: Higher-order graph neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 4602–4609, 2019.   \n[12] Sami Abu-El-Haija, Bryan Perozzi, Amol Kapoor, Nazanin Alipourfard, Kristina Lerman, Hrayr Harutyunyan, Greg Ver Steeg, and Aram Galstyan. Mixhop: Higher-order graph convolutional architectures via sparsified neighborhood mixing. In international conference on machine learning, pages 21–29. PMLR, 2019.   \n[13] Giannis Nikolentzos, George Dasoulas, and Michalis Vazirgiannis. k-hop graph neural networks. Neural Networks, 130:195–205, 2020.   \n[14] Guangtao Wang, Rex Ying, Jing Huang, and Jure Leskovec. Multi-hop attention graph neural networks. In International Joint Conference on Artificial Intelligence, 2021.   \n[15] Eli Chien, Jianhao Peng, Pan Li, and Olgica Milenkovic. Adaptive universal generalized pagerank graph neural network. In International Conference on Learning Representations, 2021. URL https://openreview.net/forum?id=n6jl7fLxrP.   \n[16] Rémy Brossard, Oriel Frigo, and David Dehaene. Graph convolutions that can finally model local structure. arXiv preprint arXiv:2011.15069, 2020.   \n[17] Chengxuan Ying, Tianle Cai, Shengjie Luo, Shuxin Zheng, Guolin Ke, Di He, Yanming Shen, and Tie-Yan Liu. Do transformers really perform badly for graph representation? In A. Beygelzimer, Y. Dauphin, P. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems, 2021. URL https://openreview.net/forum?id=OeWooOxFwDa. ",
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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes]   \n(c) Did you discuss any potential negative societal impacts of your work? [N/A]   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] ",
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+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The implementation of KP-GNN can be found at https://github.com/JiaruiFeng/KP-GNN.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] ",
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+# EFFICIENT ACTIVE SEARCH FOR COMBINATORIAL OPTIMIZATION PROBLEMS
+
+André Hottung Bielefeld University, Germany andre.hottung@uni-bielefeld.de
+
+Yeong-Dae Kwon Samsung SDS, Korea y.d.kwon@samsung.com
+
+Kevin Tierney Bielefeld University, Germany kevin.tierney@uni-bielefeld.de
+
+# ABSTRACT
+
+Recently, numerous machine learning based methods for combinatorial optimization problems have been proposed that learn to construct solutions in a sequential decision process via reinforcement learning. While these methods can be easily combined with search strategies like sampling and beam search, it is not straightforward to integrate them into a high-level search procedure offering strong search guidance. Bello et al. (2016) propose active search, which adjusts the weights of a (trained) model with respect to a single instance at test time using reinforcement learning. While active search is simple to implement, it is not competitive with state-of-the-art methods because adjusting all model weights for each test instance is very time and memory intensive. Instead of updating all model weights, we propose and evaluate three efficient active search strategies that only update a subset of parameters during the search. The proposed methods offer a simple way to significantly improve the search performance of a given model and outperform state-of-the-art machine learning based methods on combinatorial problems, even surpassing the well-known heuristic solver LKH3 on the capacitated vehicle routing problem. Finally, we show that (efficient) active search enables learned models to effectively solve instances that are much larger than those seen during training.
+
+# 1 INTRODUCTION
+
+In recent years, a wide variety of machine learning (ML) based methods for combinatorial optimization problems have been proposed (e.g., Kool et al. (2019); Hottung et al. (2020)) . While early approaches failed to outperform traditional operations research methods, the gap between handcrafted and learned heuristics has been steadily closing. However, the main potential of ML-based methods lies not only in their ability to outperform existing methods, but in automating the design of customized heuristics in situations where no handcrafted heuristics have yet been developed. We hence focus on developing approaches that require as little additional problem-specific knowledge as possible.
+
+Existing ML based methods for combinatorial optimization problems can be classified into construction methods and improvement methods. Improvement methods search the space of complete solutions by iteratively refining a given start solution. They allow for a guided exploration of the search space and are able to find high-quality solutions. However, they usually rely on problemspecific components. In contrast, construction methods create a solution sequentially starting from an empty solution (i.e., they consider a search space consisting of incomplete solutions). At test time, they can be used to either greedily construct a single solution or to sample multiple solutions from the probability distribution encoded in the trained neural network. Furthermore, the sequential solution generation process can be easily integrated into a beam search without requiring any problem-specific components. However, search methods like sampling and beam search offer no (or very limited) search guidance. Additionally, these methods do not react towards the solutions seen so far, i.e., the underlying distribution from which solutions are sampled is never changed throughout the search.
+
+Bello et al. (2016) propose a generic search strategy called active search that allows an extensive, guided search for construction methods without requiring any problem specific components. Active search is an iterative search method that at each iteration samples solutions for a single test instance using a given model and then adjusts the parameters of that model with the objective to increase the likelihood of generating high-quality solutions in future iterations. They report improved performance over random sampling when starting the search from an already trained model. Despite promising results, active search has not seen adaption in the literature. The reason for this is its resource requirements, as adjusting all model parameters separately for each test instance is very time intensive, especially compared to methods that can sample solutions to multiple different instances in one batch.
+
+We extend the idea of active search as follows. (1) We propose to only adjust a subset of (model) parameters to a single instance during the search, while keeping all other parameters fixed. We show that this efficient active search (EAS) drastically reduces the runtime of active search without impairing the solution quality. (2) We implement and evaluate three different implementations of EAS and show that all offer significantly improved performance over pure sampling approaches.
+
+In our EAS implementations, the majority of (model) parameters are not updated during the search, which drastically reduces the runtime, because gradients only need to be computed for a subset of model weights, and most operations can be applied identically across a batch of different instances. Furthermore, we show that for some problems, EAS finds even better solutions than the original active search. All EAS implementations can be easily applied to existing ML construction methods.
+
+We evaluate the proposed EAS approaches on the traveling salesperson problem (TSP), the capacitated vehicle routing problem (CVRP) and the job shop scheduling problem (JSSP). For all problems, we build upon already existing construction approaches that only offer limited search capabilities. In all experiments, EAS leads to significantly improved performance over sampling approaches. For the CVRP and the JSSP, the EAS approaches outperform all state-of-the-art ML based approaches, and even the well-known heuristic solver LKH3 for the CVRP. Furthermore, EAS approaches assists in model generalization, resulting in drastically improved performance when searching for solutions to instances that are much larger than the instances seen during model training.
+
+# 2 LITERATURE REVIEW
+
+Construction methods Hopfield (1982) first used a neural network (a Hopfield network) to solve small TSP instances with up to 30 cities. The development of recent neural network architectures has paved the way for ML approaches that are able to solve large instances. The pointer network architecture proposed by Vinyals et al. (2015) efficiently learns the conditional probability of a permutation of a given input sequence, e.g., a permutation of cities for a TSP solution. The authors solve TSP instances with up to 50 cities via supervised learning. Bello et al. (2016) report that training a pointer network via actor-critic RL instead results in a better performance on TSP instances with 50 and 100 cities. Furthermore, graph neural networks are used to solve the TSP, e.g., a graph embedding network in Khalil et al. (2017) and a graph attention network in Deudon et al. (2018).
+
+The first applications of neural network based methods to the CVRP are reported by Nazari et al. (2018) and Kool et al. (2019). Nazari et al. (2018) propose a model with an attention mechanism and a recurrent neural network (RNN) decoder that can be trained via actor-critic RL. Kool et al. (2019) propose an attention model that uses an encoder that is similar to the encoder used in the transformer architecture Vaswani et al. (2017). Peng et al. (2019) and Xin et al. (2021) extend the attention model to update the node embeddings throughout the search, resulting in improved performance at the cost of longer runtimes for the CVRP. Falkner & Schmidt-Thieme (2020) propose an attention-based model that constructs tours in parallel for the CVRP with time windows.
+
+While ML-based construction methods have mainly focused on routing problems, there are some notable exceptions. For example, Khalil et al. (2017) use a graph embedding network approach to solve the minimum vertex cover and the maximum cut problems (in addition to the TSP). Zhang et al. (2020) propose a graph neural network based approach for the job shop scheduling problem (JSSP). Li et al. (2018) use a guided tree search enhanced ML approach to solve the maximal independent set, minimum vertex cover, and the maximal clique problems. For a more detailed review of ML methods on different combinatorial optimization problems, we refer to Vesselinova et al. (2020).
+
+While most approaches construct routing problem solutions autoregressively, some approaches predict a heat-map that describes which edges will likely be part of a good solution. The heat-map is then used in a post-hoc search to construct solutions. Joshi et al. (2019) use a graph convolutional network to create a heat-map and a beam search to search for solutions. Similarly, Fu et al. (2020) use a graph convolutional residual network with Monte Carlo tree search to solve large TSP instances. Kool et al. (2021) use the model from Joshi et al. (2019) to generate the heat-map and use it to search for solution to TSP and CVRP instances with a dynamic programming based approach.
+
+Improvement methods Improvement methods integrate ML based methods into high-level search heuristics or try to learn improvement operators directly. In general, they often invest more time into solving an instance than construction based methods (and usually find better solutions). Chen & Tian (2019) propose an approach that iteratively changes a local part of the solution. At each iteration, the trainable region picking policy selects a part of the solution that should be changed and a trainable rule picking policy selects an action from a given set of possible modification operations. Hottung & Tierney (2020) propose a method for the CVRP that iteratively destroys parts of a solution using predefined, handcrafted operators and then reconstructs them with a learned repair operator. Wu et al. (2021) and de O. da Costa et al. (2020) propose to use RL to pick an improving solution from a specified local neighborhood (e.g., the 2-Opt neighborhood) to solve routing problems. Hottung et al. (2021) learn a continuous representation of discrete routing problem solutions using conditional variational autoencoders and search for solutions using a generic, continuous optimizer.
+
+# 3 SOLVING COMBINATORIAL OPTIMIZATION PROBLEMS WITH EAS
+
+We propose three EAS implementations that adjust a small subset of (model) parameters in an iterative search process. Given an already trained model, we investigate adjusting (1) the normally static embeddings of the problem instance that are generated by the encoder model, (2) the weights of additional instance-specific residual layers added to the decoder, and (3) the parameters of a lookup table that directly affect the probability distribution returned by model. In each iteration, multiple solutions are sampled for one instance and the dynamic (model) parameters are adjusted with the goal of increasing the probability of generating high quality solutions (as during model training). This allows the search to sample solutions of higher quality in subsequent iterations, i.e., the search can focus on the more promising areas of the search space. Once a high-quality solution for an instance is found, the adjusted parameters are discarded, so that the search process can be repeated on other instances. All strategies efficiently generate solutions to a batch of instances in parallel, because the network layers not updated during the search are applied identically to all instances of the batch.
+
+Background RL based approaches for combinatorial problems aim to learn a neural network based model $p _ { \theta } ( \pi | l )$ with weights $\theta$ that can be used to generate a solution $\pi$ given an instance l. State-ofthe-art approaches usually use a model that consists of an encoder and a decoder unit. The encoder usually creates static embeddings $\omega$ that describe the instance $l$ using a computationally expensive encoding process (e.g., Kool et al. (2019); Kwon et al. (2020)). The static embeddings are then used to autoregessively construct solutions using the decoder over $T$ time steps. At each step $t$ , the decoder $q _ { \phi } ( a | s _ { t } , \bar { \omega } )$ , with weights $\phi \subset \theta$ , outputs a probability value for each possible action $a$ in the state $s _ { t }$ (e.g., for the TSP, each action corresponds to visiting a different city next). The starting state $s _ { 1 }$ describes the problem instance $l$ (e.g., the positions of the cities for the TSP and the starting city) and the state $s _ { t + 1 }$ is obtained by applying the action $a _ { t }$ selected at time step $t$ to the state $s _ { t }$ . The (partial) solution $\pi _ { t }$ is defined by the sequence of selected actions $a _ { 1 } , a _ { 2 } , \ldots , a _ { t }$ . Once a complete solution, $\pi _ { T }$ , fulfilling all constraints of the problem is constructed, the objective function value $\mathbf { \bar { \boldsymbol { C } } } ( \pi , l )$ of the solution can be computed (e.g., the tour length for the TSP).
+
+Figure 1 shows the solution generation for a TSP instance with a model that uses static embeddings. The static embeddings $\omega$ are used at each decoding step to generate a probability distribution over all possible next actions and the selected action is provided to the decoder in the next decoding step. During testing, solutions can be constructed by either selecting actions greedily or by sampling each action according to $q _ { \phi } ( a | s _ { t } , \omega )$ . Since the static embeddings are not updated during solution generation they only need to be computed once per instance, which allows to quickly sample multiple solutions per instance. We note that not all models use static embeddings. Some approaches update all instance embeddings after each action (e.g., Zhang et al. (2020)), which allows the embeddings to contain information on the current solution state $s _ { t }$ .
+
+![](images/3abde8df1e97587e52664ac4987fda3a29e2025873cd251133e478eda9740eb0.jpg)  
+Figure 1: Sampling a solution for the TSP with a model $p _ { \theta } ( \pi | l )$ that uses static instance embeddings.
+
+# 3.1 EMBEDDING UPDATES
+
+Our first proposed strategy, called EAS-Emb, updates the embeddings $\omega$ generated by an encoder using a loss function consisting of an RL component $\mathcal { L } _ { R L }$ and an imitation learning $\left( \operatorname { I L } \right)$ component $\mathcal { L } _ { I L }$ . The loss $\mathcal { L } _ { R L }$ is based on REINFORCE (Williams, 1992) and is the expected cost of the generated solutions, $\mathbb { E } \left[ C ( \pi ) \right]$ . We aim to adjust the embedding parameters to increase the likelihood of generating solutions with lower costs (e.g., a shorter tour length for the TSP). The loss $\mathcal { L } _ { I L }$ is the negation of the log-probability of (re-)generating the best solution seen so far. We adjust the embedding parameters to increase this probability.
+
+More formally, for an instance $l$ we generate the embeddings $\omega$ using a given encoder. Based on $\omega$ we can (repeatedly) sample a solution $\pi$ whose cost is $C ( \pi )$ . A subset of the embeddings $\hat { \omega } \subseteq \omega$ is adjusted to minimize $\mathcal { L } _ { R L }$ using the gradient
+
+$$
+\nabla _ { \hat { \omega } } \mathcal { L } _ { R L } ( \hat { \omega } ) = \mathbb { E } _ { \boldsymbol \pi } \left[ ( C ( \boldsymbol \pi ) - b _ { \circ } ) \nabla _ { \hat { \omega } } \log q _ { \phi } ( \boldsymbol \pi \mid \hat { \omega } ) \right]
+$$
+
+where $\begin{array} { r } { q _ { \phi } ( \pi \mid \hat { \omega } ) \equiv \prod _ { t = 1 } ^ { T } q _ { \phi } ( a _ { t } \mid s _ { t } , \hat { \omega } ) } \end{array}$ , and $b _ { \circ }$ is a baseline (we use the baseline proposed in Kwon et al. (2020) for our experiments).
+
+For the second loss $\mathcal { L } _ { I L }$ , let $\bar { \pi }$ be the best solution found so far for the instance $l$ , that consists of the actions $\bar { a } _ { 1 } , \dots , \bar { a } _ { T }$ . We use teacher forcing to make the decoder $q _ { \phi } ( \cdot | s _ { t } , \hat { \omega } )$ generate the solution $\bar { \pi }$ , during which we obtain the probability values associated with the actions $\bar { a } _ { 1 } , \dots , \bar { a } _ { T }$ . We increase the log-likelihood of generating $\bar { \pi }$ by adjusting $\hat { \omega }$ using the gradient
+
+$$
+\nabla _ { \boldsymbol { \hat { \omega } } } \mathcal { L } _ { I L } ( \boldsymbol { \hat { \omega } } ) = - \nabla _ { \boldsymbol { \hat { \omega } } } \log q _ { \phi } ( \bar { \pi } \mid \boldsymbol { \hat { \omega } } ) \equiv - \nabla _ { \boldsymbol { \hat { \omega } } } \log \prod _ { t = 1 } ^ { T } q _ { \phi } ( \bar { a } _ { t } | s _ { t } , \boldsymbol { \hat { \omega } } ) .
+$$
+
+The gradient of the overall loss $\mathcal { L } _ { R I L }$ is defined as $\nabla _ { \hat { \omega } } \mathcal { L } _ { R I L } ( \hat { \omega } ) = \nabla _ { \hat { \omega } } \mathcal { L } _ { R L } ( \hat { \omega } ) + \lambda \cdot \nabla _ { \hat { \omega } } \mathcal { L } _ { I L } ( \hat { \omega } )$ , where $\lambda$ is a tunable parameter. If a high value for $\lambda$ is selected, the search focuses on generating solutions that are similar to the incumbent solution. This accelerates the convergence of the search policy, which is useful when the number of search iterations is limited.
+
+We note that both decoding processes required for RL and $\mathrm { I L }$ can be carried out in parallel, using the same forward pass through the network. Furthermore, only the parameters $\hat { \omega }$ are instance specific, while all other model parameters are identical for all instances. This makes parallelization of multiple instances in a batch more efficient both in time and memory.
+
+# 3.2 ADDED-LAYER UPDATES
+
+We next propose EAS-Lay, which adds an instance-specific residual layer to a trained model. During the search, the weights in the added layer are updated, while the weights of all other original layers are held fixed. We use both RL and $\mathrm { I L }$ , similarly to EAS-Emb in Section 3.1.
+
+We formalize EAS-Lay as follows. For each instance $l$ we insert a layer
+
+$$
+\operatorname { L } ^ { \star } ( h ) = h + ( ( \operatorname { R e L u } ( h W ^ { 1 } + b ^ { 1 } ) W ^ { 2 } + b ^ { 2 } )
+$$
+
+into the given decoder $q _ { \phi }$ , resulting in a slightly modified model $\tilde { q } _ { \phi , \psi }$ , where $\psi = \{ W ^ { 1 } , b ^ { 1 } , W ^ { 2 } , b ^ { 2 } \}$ . The layer takes in the input $h$ and applies two linear transformations with a ReLu activation function in between. The weight matrices $W ^ { \bar { 1 } }$ and $W ^ { 2 }$ and the bias vectors $b ^ { 1 }$ and $b ^ { 2 }$ are adjusted throughout the search via gradient descent. The weights in the matrix $W ^ { 2 }$ and the vector $b ^ { 2 }$ are initialized to zero so that the added layer does not affect the output of the model during the first iteration of the search. The gradient for $\mathcal { L } _ { R L }$ is given as
+
+$$
+\nabla _ { \psi } \mathcal { L } _ { R L } ( \psi ) = \mathbb { E } _ { \pi } \big [ ( C ( \pi ) - b _ { \circ } ) \nabla _ { \psi } \log \tilde { q } _ { \phi , \psi } ( \pi ) \big ] ,
+$$
+
+with $\begin{array} { r } { \tilde { q } _ { \phi , \psi } ( \pi ) \equiv \prod _ { t = 1 } ^ { T } \tilde { q } _ { \phi , \psi } ( a _ { t } \mid s _ { t } , \omega ) } \end{array}$ , and $b _ { \circ }$ is a baseline. The gradient for $\mathcal { L } _ { I L }$ is defined similarly.
+
+Note that the majority of the network operations are not instance specific. They can be applied identically to all instances running in parallel as a batch, resulting in significantly lower runtime during search. The position at which the new layer is inserted has an impact on the performance of EAS-Lay, and identifying the best position usually requires testing. In general, the memory requirement of this approach can be reduced by inserting the additional layer closer towards the output layer of the network. This decreases the number of layers to be considered during backpropagation. We noticed for transformer-based architectures that applying the residual layer $\mathrm { L } ^ { \star } ( \cdot )$ to the query vector $q$ before it is passed to the single attention head usually results in a good performance.
+
+# 3.3 TABULAR UPDATES
+
+EAS-Emb and EAS-Lay require significantly less memory per instance than the original active search. However, they still need to store many gradient weights associated with multiple layers for the purpose of backpropagation. This significantly limits the number of solutions one can generate in parallel. We hence propose EAS-Tab, which does not require backpropagation, but instead uses a simple lookup table to modify the policy of the given model. For each action at a given state, the table provides a guide on how to change its probability, so that the sampled solution has a higher chance at being similar to the best solution found in the past.
+
+Formally, at each step $t$ during the sequential generation of a solution, we redefine the probability of selecting action $a _ { t }$ in the state $s _ { t }$ as $q _ { \phi } ( a | \bar { s _ { t } } , \omega ) ^ { \alpha } \cdot Q _ { g ( s _ { t } , a _ { t } ) }$ and renormalize over all possible actions using the softmax function. Here, $\alpha$ is a hyperparameter, and $g$ is a function that maps each possible state and action pair to an entry in the table $Q$ . The network parameters $\theta$ remain unchanged, resulting in fast and memory efficient solution generation. The hyperparameter $\alpha$ is similar to the temperature value proposed in Bello et al. (2016) and modifies the steepness of the probability distribution returned by the model (lower values increase the exploration of the search). During search, the table $Q$ is updated with the objective of increasing the quality of the generated solutions. More precisely, after each iteration, $Q$ is updated based on the best solution $\bar { \pi }$ found so far consisting of the actions $\bar { a } _ { 1 } , \dots , \bar { a } _ { T }$ at states $\bar { s } _ { 1 } , \dots , \bar { s } _ { T }$ , respectively, with
+
+$$
+Q _ { g ( s _ { t } , a _ { t } ) } = \left\{ \begin{array} { l l } { \operatorname* { m a x } ( 1 , \frac { \sigma } { q _ { \phi } ( a | s _ { t } , \omega ) ^ { \alpha } } ) , } & { \mathrm { i f } g ( s _ { t } , a _ { t } ) \in \{ g ( \bar { a } _ { 1 } , \bar { s } _ { 1 } ) , \dots , g ( \bar { a } _ { T } , \bar { s } _ { T } ) \} } \\ { 1 , } & { \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+
+The hyperparameter $\sigma$ defines the degree of exploitation of the search. If a higher value of $\sigma$ is used, the probabilities for actions that generate the incumbent solution are increased.
+
+In contrast to embedding or added-layer updates, this EAS method requires deeper understanding of the addressed combinatorial optimization problem to design the function $g ( s _ { t } , a _ { t } )$ . For example, for the TSP with $n$ nodes we use a table $Q$ of size $n \times n$ in which each entry $Q _ { i , j }$ corresponds to a directed edge $e _ { i , j }$ of the problem instance. The probability increases for the same directed edge that was used in the incumbent solution. This definition of $g ( s _ { t } , a _ { t } )$ effectively ignores the information on all the previous visits stored in state $s _ { t }$ , focusing instead on the current location (city) in choosing the next move. We note that this EAS approach is similar to the ant colony optimization algorithm (Dorigo et al., 2006), which has been applied to a wide variety of combinatorial optimization problems.
+
+# 4 EXPERIMENTS
+
+We evaluate all EAS strategies using existing, state-of-the-art RL based methods for three different combinatorial optimization problems. For the first two, the TSP and the CVRP, we implement EAS for the POMO approach (Kwon et al., 2020). For the third problem, the JSSP, we use the L2D method from Zhang et al. (2020). We extend the code made available by the authors of POMO (MIT license) and L2D (no license) with our EAS strategies to ensure a fair evaluation. Note that we only make minor modifications to these methods, and we use the models trained by the authors when available.
+
+We run all experiments on a GPU cluster using a single Nvidia Tesla V100 GPU and a single core of an Intel Xeon 4114 CPU at $2 . 2 \ : \mathrm { G H z }$ for each experiment. Our source code is available at https://github.com/ahottung/EAS. We use the Adam optimizer (Kingma & Ba, 2014) for all EAS approaches. The hyperparameters $\lambda , \sigma , \alpha$ , and the learning rate for the optimizer are tuned via Bayesian optimization using scikit-optimize (Head et al., 2020) on separate validation instances, which are sampled from the same distribution as the test instances. The hyperparameters are not adjusted for larger instances used to evaluate the generalization performance.
+
+# 4.1 TSP
+
+The TSP is a well-known routing problem involving finding the shortest tour between a set of $n$ nodes (i.e., cities) that visits each node exactly once and returns to the starting node. We assume that the distance matrix obeys the triangle inequality.
+
+Implementation POMO uses a model that is very similar to the AM model from Kool et al. (2019). The model generates instance embeddings only once per instance and does not update them during construction. The probability distribution over all actions are generated by a decoder, whose last layer is a single-headed attention layer. This last layer calculates the compatibility of a query vector $q$ to the key vector $k _ { i }$ for each node $i$ . In this operation, the key vector $k _ { i }$ is an embedding that has been computed separately, but identically for each input (i.e., node $i$ ) during the instance encoding process. For EAS-Emb, we only update the set of single-head keys $k _ { i }$ $( i = 1 , \ldots , n )$ . For EAS-Lay we apply the residual layer $\mathrm { L } ^ { \star } ( \cdot )$ described in Equation 3 to the query vector $q$ before it is passed to the single attention head. For EAS-Tab, we use a table $Q$ of size $n \times n$ and the mapping function $g ( s _ { t } , a _ { t } )$ such that each entry $Q _ { i , j }$ corespondents to a directed edge $e _ { i , j }$ of the problem instance.
+
+Setup We use the 10,000 TSP instances with $n = 1 0 0$ from Kool et al. (2019) for testing and three additional sets of 1,000 instances to evaluate generalization performance. We evaluate the EAS approaches against just using POMO with greedy action selection, random sampling, and active search as in Bello et al. (2016). In all cases, we use the model trained on instances with $n = 1 0 0$ made available by the POMO authors. For greedy action selection, POMO generates $8 \cdot n$ solutions for an instance of size $n$ (using 8 augmentations and $n$ different starting cities). In all other cases, we generate $2 0 0 \cdot 8 \cdot n$ solutions per instance (over the course of 200 iterations for the (E)AS approaches). The batch size (the number of instances solved in parallel) is selected for each method individually to fully utilize the available GPU memory. We compare to the exact solver Concorde (Applegate et al., 2006), the heuristic solver LKH3 (Helsgaun, 2017), the graph convolutional neural network with beam search (GCN-BS) from Joshi et al. (2019), the 2-Opt based deep learning (2-Opt-DL) approach from de O. da Costa et al. (2020), the learning improvement heuristics (LIH) method from Wu et al. (2021), the conditional variational autoencoder (CVAE-Opt) approach (Hottung et al., 2021), and deep policy dynamic programming (DPDP) (Kool et al., 2021).
+
+Results Table 1 shows the average costs, average gap and the total runtime (wall-clock time) for each instance set. The exact solver Concorde performs best overall, as it is a highly specialized TSP solver. Of the POMO-based approaches, the original active search offers the best gap to optimality, but requires 5 days of runtime. EAS significantly lowers the runtime while the gap is only marginally larger. DPDP performs best among ML-based approaches. However, DPDP relies on a handcrafted and problem-specific beam search, whereas EAS methods are completely problem-independent. On the larger instances, EAS significantly improves generalization performance, reducing the gap over sampling by up to $3 . 6 \mathrm { x }$ . We also evaluate active search using the imitation learning loss, but observe no impact on the search performance (see Appendix B).
+
+Table 1: Results for the TSP   
+
+<table><tr><td rowspan="2"></td><td colspan="3">Testing (10k inst.)</td><td colspan="10">Generalization (1k instances)</td></tr><tr><td colspan="3">n=100</td><td colspan="3">n=125</td><td colspan="3">n=150</td><td colspan="3"></td><td>n = 200</td></tr><tr><td>Method</td><td>Obj.</td><td>Gap</td><td>Time</td><td>Obj.</td><td>Gap</td><td>Time</td><td>Obj.</td><td>Gap</td><td></td><td>Time</td><td>Obj.</td><td>Gap</td><td>Time</td></tr><tr><td>Concorde</td><td></td><td>[7.765 0.000%</td><td></td><td></td><td>82M|8.583 0.000%</td><td></td><td></td><td>12M|9.346 0.000%</td><td></td><td></td><td>17M|10.687 0.000%</td><td></td><td>31M</td></tr><tr><td>LKH3</td><td>7.765</td><td>0.000%</td><td>8H</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>8.5830.000%73M9.3460.000% 99M10.6870.000%</td><td>3H</td></tr><tr><td>GCN-BS</td><td>7.87</td><td>1.39%</td><td>40M</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td><td>-</td></tr><tr><td>2-Opt-DL</td><td>7.83</td><td>0.87%</td><td>41M</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td></tr><tr><td>LIH</td><td>7.87</td><td>1.42%</td><td>2H</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td></tr><tr><td>CVAE-Opt</td><td>1</td><td>0.343%</td><td>6D</td><td></td><td>8.646 0.736%</td><td>21H</td><td></td><td>9.482 1.454%</td><td></td><td>30H</td><td></td><td></td><td></td></tr><tr><td>DPDP</td><td>7.765</td><td>0.004%</td><td>2H</td><td></td><td>8.589 0.070%</td><td>31M</td><td></td><td>9.4340.942%</td><td></td><td>44M</td><td></td><td>11.154 4.370%</td><td>74M</td></tr><tr><td>Greedy</td><td>7.776</td><td>0.146%</td><td>1M|</td><td>8.607</td><td>0.278%</td><td></td><td>&lt;1M|9.397</td><td></td><td>0.542%</td><td>&lt;1M|</td><td>10.843</td><td>1.457%</td><td>1M</td></tr><tr><td>Sampling</td><td>7.770</td><td>0.074%</td><td>4H</td><td>8.595</td><td>0.145%</td><td>45M</td><td></td><td>9.378</td><td>0.334%</td><td>78M</td><td></td><td>10.8381.416%</td><td>3H</td></tr><tr><td>Active S.</td><td></td><td>7.768 0.046%</td><td>5D</td><td>8.591</td><td>0.095%</td><td>15H</td><td>9.364</td><td></td><td>0.192%</td><td>19H</td><td>10.735</td><td>50.447%</td><td>24H</td></tr><tr><td>0 EAS-Emb</td><td></td><td>7.769 0.063%</td><td>5H</td><td>8.591</td><td>0.092%</td><td>57M</td><td></td><td>9.363</td><td>0.174%</td><td>2H</td><td>10.730 0.400%</td><td></td><td>4H</td></tr><tr><td>P EAS-Lay</td><td></td><td>7.769 0.053%</td><td>7H</td><td>8.591</td><td>0.089%</td><td>74M</td><td></td><td>9.363 0.176%</td><td></td><td>2H</td><td>10.737 0.471%</td><td></td><td>4H</td></tr><tr><td>EAS-Tab</td><td></td><td>7.768 0.048%</td><td>5H</td><td>8.591</td><td>0.091%49M</td><td></td><td></td><td>9.3650.196%</td><td></td><td>1H</td><td>10.756 0.650%</td><td></td><td>3H</td></tr></table>
+
+# 4.2 CVRP
+
+The goal of the CVRP is to find the shortest routes for a set of vehicles with limited capacity that must deliver goods to a set of $n$ customers. We again use the POMO approach as a basis for our EAS strategies. As is standard in the ML literature, we evaluate all approaches on instance sets where the locations and demands are sampled uniformly at random. Additionally, we consider the more realistic instance sets proposed in Hottung & Tierney (2020) with up to 297 customers (see Appendix A).
+
+Implementation We use the same EAS implementation for the CVRP as for the TSP.
+
+Setup We use the 10,000 CVRP instances from Kool et al. (2019) for testing and additional sets of 1,000 instances to evaluate the generalization performance. Again, we compare the EAS approaches to POMO using greedy action selection, sampling and active search. We generate the same number of solutions per instance as for the TSP. We compare to LIH, CAVE-Opt, DPDP, NeuRewriter (Chen & Tian, 2019) and neural large neighborhood search (NLNS) from Hottung & Tierney (2020).
+
+Results Table 2 shows the average costs, the average gap to LKH3 and the total wall-clock time for all instance sets. EAS-Lay outperforms all other approaches on the test instances, including approaches that rely on problem-specific knowledge, with a gap that beats LKH3. Both other EAS methods also find solutions of better quality than LKH3, which is quite an accomplishment given the many years of work on the LKH3 approach. We note it is difficult to provide a fair comparison between a single-core, CPU-bound technique like LKH3 and our approaches that use a GPU. Nonetheless, assuming a linear speedup, at least 18 CPU cores would be needed for LKH3 to match the runtime of EAS-Tab. On the generalization instance sets with $n = 1 2 5$ and $n = 1 5 0$ , the EAS approaches also outperform LKH3 and CVAE-Opt while being significantly faster than active search. On the instances with $n = 2 0 0$ , active search finds the best solutions of all POMO based approaches with a gap of $0 . 2 2 \%$ to LKH3, albeit with a long runtime of 36 hours. We hypothesize that significant changes to the learned policy are necessary to generate high-quality solutions for instances that are very different to those seen during training. Active search’s ability to modify all model parameters makes it easier to make those changes. EAS-Tab offers the worst performance on the instances with $n = 2 0 0$ with a gap of $1 1 . 8 \%$ . This is because EAS-Tab is very sensitive to the selection of the hyperparameter $\alpha$ , meaning that EAS-Tab requires hyperparameter tuning on some problems to generalize more effectively. Adjusting $\alpha$ for the $n = 2 0 0$ case improves EAS-Tab’s gap to at least $3 . 5 4 \%$ , making it slightly better than greedy or sampling.
+
+Table 2: Results for the CVRP on instances with uniformly sampled locations and demands   
+
+<table><tr><td rowspan="2"></td><td colspan="3">Testing (10k inst.) n=100</td><td colspan="10">Generalization (1k instances)</td></tr><tr><td>Obj.</td><td>Gap</td><td>Time</td><td>Obj.</td><td>n =125 Gap</td><td>Time</td><td>Obj.</td><td>n=150 Gap</td><td>Time</td><td>Obj.</td><td>n = 200 Gap</td><td></td><td>Time</td></tr><tr><td>Method</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>LKH3</td><td>|15.65</td><td>0.00%</td><td></td><td>6D|17.50</td><td>0.00%</td><td></td><td>19H|19.22</td><td></td><td>0.00%</td><td></td><td>20H|22.00</td><td>0.00%</td><td>25H</td></tr><tr><td>NLNS NeuRewriter</td><td>15.99</td><td>2.23%</td><td>62M</td><td>[|18.07</td><td>3.23%</td><td></td><td>9M|19.96</td><td></td><td>3.86%</td><td>12M|23.02</td><td></td><td>4.66%</td><td>24M</td></tr><tr><td>LIH</td><td>16.10</td><td></td><td>66M</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td></tr><tr><td>CVAE-Opt</td><td>16.03</td><td>2.47% 1.36%</td><td>5H</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td><td>=</td></tr><tr><td>DPDP</td><td>15.63</td><td>-0.13%</td><td>11D 23H</td><td>17.87 17.51</td><td>2.08% 0.07%</td><td>36H 3H</td><td>19.84 19.31</td><td></td><td>3.24% 0.48%</td><td>46H 5H</td><td>22.26</td><td>51.20%</td><td>= 9H</td></tr><tr><td>Greedy</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>15.76</td><td>0.76%</td><td>2M</td><td>17.73</td><td>1.29%</td><td>&lt;1M</td><td>19.64</td><td></td><td>2.18%</td><td>1M</td><td>|22.90</td><td>4.12%</td><td>1M</td></tr><tr><td>Sampling 0</td><td>15.67</td><td>0.17%</td><td>7H</td><td>17.60</td><td>0.54%</td><td>73M</td><td>19.48</td><td></td><td>1.35%</td><td>2H</td><td>23.18</td><td>5.35%</td><td>5H</td></tr><tr><td>Active S.</td><td>15.63</td><td>-0.07%</td><td>8D</td><td>17.47</td><td>-0.21%</td><td>25H</td><td>19.21</td><td></td><td>-0.03%</td><td>29H</td><td>22.05</td><td>0.22%</td><td>36H</td></tr><tr><td>EAS-Emb</td><td>15.63 15.61</td><td>-0.08%</td><td>9H 12H</td><td>17.47</td><td>-0.21%</td><td>93M</td><td>19.22</td><td></td><td>0.03%</td><td>3H</td><td>22.19</td><td>0.88%</td><td>6H</td></tr><tr><td>EAS-Lay EAS-Tab</td><td>15.62</td><td>-0.23% -0.14%</td><td>8H</td><td>17.50</td><td>17.46 -0.24% 0.00%</td><td>2H 80M</td><td>19.21 19.36</td><td></td><td>-0.04% 0.72%</td><td>3H 2H</td><td>22.10 24.56</td><td>0.45% 11.8%</td><td>8H 5H</td></tr></table>
+
+# 4.3 JSSP
+
+The JSSP is a scheduling problem involving assigning jobs to a set of heterogeneous machines. Each job consists of multiple operations that are run sequentially on the set of machines. The objective is to minimize the time needed to complete all jobs, called the makespan. We evaluate EAS using the L2D approach, which is a state-of-the-art ML based construction method using a graph neural network.
+
+Implementation L2D represents JSSP instances as disjunctive graphs in which each operation of an instance is represented by a node in the graph. To create a schedule, L2D sequentially selects the operation that should be scheduled next. To this end, an embedding $h _ { v }$ is created for each node $v$ in a step-wise encoding process. In contrast to POMO, the embeddings $h _ { v }$ are recomputed after each decision step $t$ . Since EAS-Emb requires static embeddings, we modify the network to use $\tilde { h } _ { v } ^ { t } = h _ { v } ^ { t } + h _ { v } ^ { S T }$ as an embedding for node t to zero. During the searc $v$ at step  with E $t$ , where S-Emb $h _ { v } ^ { S T }$ is a vector that is initialized winly adjust the static component $h _ { v } ^ { S T }$ of the embedding with gradient descent. For EAS-Lay, we insert the residual layer $\mathrm { L } ^ { \star } ( \cdot )$ described in Equation 3 to each embedding $h _ { v }$ separately and identically. Finally, for EAS-Tab, we use a table $Q$ of size $| O | \times | O |$ , where $| O |$ is the number of operations, and we design the function $g ( s _ { t } , a _ { t } )$ so that the entry $Q _ { i , j }$ corresponds to selecting the operation $o _ { j }$ directly after the operation $o _ { i }$ .
+
+Setup We use three instance sets with 100 instances from Zhang et al. (2020) for testing and to evaluate the generalization performance. We use the exact solver Google OR-Tools (Perron & Furnon) as a baseline, allowing it a maximum runtime of 1 hour per instance. Furthermore, we compare to L2D with greedy action selection. Note that the performance of the L2D implementation is CPU bound and does not allow different instances to be batch processed. We hence solve instances sequentially and generate significantly fewer solutions per instance than for the TSP and the CVRP. For sampling, active search and the EAS approaches we sample 8,000 solutions per problem instance over the course of 200 iterations for the (efficient) active search approaches.
+
+Results Table 3 shows the average gap to the OR-Tools solution and the total wall-clock time per instance set. EAS-Emb offers the best performance for all three instance sets. On the $1 0 \times 1 0$ instances, EAS-Emb reduces the gap by $50 \%$ in comparison to pure sampling. Even on the $2 0 \times 1 5$ instances it reduces the gap to $1 6 . 8 \%$ from $2 0 . 8 \%$ for pure sampling, despite the low number of sampled solutions per instance. EAS-Lay offers performance that is comparable to active search. We note that if L2D were to more heavily use the GPU, instances could be solved in batches, thus drastically reducing the runtime of EAS-Lay and EAS-Tab. While EAS-Tab shows similar performance to active search on the test instance set, it is unable to generalize effectively to the larger instances.
+
+# 4.4 SEARCH TRAJECTORY ANALYSIS
+
+To get a better understanding of how efficient active search improves performance, we monitor the quality of the solutions sampled at each of the 200 iterations of the search. Figure 2 reports the average quality over all test instances for the JSSP and over the first 1,000 test instances for the TSP and CVRP. As expected, the quality of solutions generated via pure sampling does not change over the course of the search for all three problems. For all other methods, the quality of the generated solutions improves throughout the search. Thus, all active search variants successfully modify the (model) parameters in a way that increases the likelihood of generating high-quality solutions.
+
+Table 3: Results for the JSSP   
+
+<table><tr><td colspan="2" rowspan="2"></td><td colspan="3">Testing(100 inst.）</td><td colspan="4">Generalization (1OO instances)</td></tr><tr><td colspan="2">10×10</td><td colspan="3">15×15</td><td colspan="3">20×15</td></tr><tr><td colspan="2" rowspan="2">Method</td><td rowspan="2">Obj. Gap</td><td rowspan="2">Time</td><td rowspan="2">Obj.</td><td>Gap</td><td>Time</td><td>Obj. Gap</td><td>Time</td></tr><tr><td rowspan="2"></td><td rowspan="2"></td><td rowspan="2"></td><td rowspan="2">0.0%</td></tr><tr><td rowspan="2">OR-Tools Greedy</td><td rowspan="2">1807.6 0.0%</td></tr><tr><td>37S|1</td><td rowspan="2">[1188.0</td><td colspan="2" rowspan="2">0.0%</td><td rowspan="2">3H| |1345.5</td><td rowspan="2"></td><td rowspan="2">80H</td></tr><tr><td rowspan="2"></td><td rowspan="2">1988.6</td></tr><tr><td>22.3% 871.7</td><td>20S 1528.3</td><td>28.6%</td><td>44S</td><td>1738.0</td><td>29.2%</td><td>60S</td></tr><tr><td rowspan="4">Sampling L</td><td>854.2</td><td>8.0% 5.8%</td><td>8H 8H</td><td>1378.3 16.0% 1345.2 13.2%</td><td>25H 32H</td><td>1624.6 1576.5</td><td>20.8% 17.2%</td><td>40H</td></tr><tr><td>EAS-Emb</td><td>837.0 3.7%</td><td>7H</td><td>1326.4 11.7%</td><td>22H</td><td>1570.8</td><td>16.8%</td><td>50H 37H</td></tr><tr><td>EAS-Lay</td><td>859.6 6.5%</td><td>7H</td><td>1352.6</td><td>13.8%</td><td>25H 1581.8</td><td>17.6%</td><td>46H</td></tr><tr><td>EAS-Tab</td><td>860.2 6.5%</td><td>8H</td><td>1376.8</td><td>15.9% 29H</td><td></td><td>1623.420.7%</td><td>51H</td></tr></table>
+
+![](images/aeff2be7dc644f08b6349a6f6cabfb88e1ed829fcc964cb99457f18da9f01b79.jpg)  
+Figure 2: Average costs of sampled solutions at each iteration (best viewed in color).
+
+![](images/3814f29ab52dc4112005668d02a29044fe9d34e4442175ae8284d12c9a420a11.jpg)  
+Figure 3: Influence of $\lambda$ on the solution quality for EAS-Emb and EAS-Lay.
+
+For the TSP, EAS-Emb and EAS-Lay offer nearly identical performance, with EAS-Tab outperforming both by a very slight margin. The original active search is significantly more unstable, which is likely the result of the learning rate being too high. Note that the learning rate has been tuned on an independent validation set. These results indicate that selecting a suitable learning rate is significantly more difficult for the original active search than for our efficient active search variants where only a subset of (model) parameters are changed. For the CVRP, all EAS variants find better solutions on average than the original search after only a few iterations. Keeping most parameters fixed seems to simplify the underlying learning problem and allows for faster convergence. For the JSSP, EAS-Emb offers significantly better performance than all other methods. The reason for this is that the L2D approach uses only two node features and has a complex node embedding generation procedure. While the original active search must fine tune the entire embedding generation process to modify the generated solutions, EAS-Emb can just modify the node embedding directly.
+
+# 4.5 ABLATION STUDY: IMITATION LEARNING LOSS
+
+We evaluate the impact of the imitation learning loss $\mathcal { L } _ { I L }$ of EAS-Emb and EAS-Lay with a sensitivity and ablation analysis for the hyperparameter $\lambda$ . We solve the first 500 test instances (to reduce the computational costs) for the TSP and CVRP, and all test instances for the JSSP using EAS-Emb and EAS-Lay with different $\lambda$ values. The learning rate remains fixed to a value determined in independent tuning runs in which $\lambda$ is fixed to zero. Figure 3 shows the results for all three problems. For the TSP and the CVRP, the results show that $\mathcal { L } _ { I L }$ can significantly improve performance. When $\lambda$ is set to 0 or very small values, $\mathcal { L } _ { I L }$ is disabled, thus including $\mathcal { L } _ { I L }$ is clearly beneficial on the TSP and CVRP. For the JSSP, the inclusion of $\mathcal { L } _ { I L }$ does not greatly improve performance, but it does not hurt it, either. Naturally, $\lambda$ should not be selected too low or too high as either too little or too much intensification can hurt search performance.
+
+# 5 CONCLUSION
+
+We presented a simple technique that can be used to extend ML-based construction heuristics by an extensive search. Our proposed modification of active search fine tunes a small subset of (model) parameters to a single instance at test time. We evaluate three example implementations of EAS that all result in significantly improved model performance in both testing and generalization experiments on three different, difficult combinatorial optimization problems. Our approach of course comes with some key limitations. Search requires time, thus for applications needing extremely fast (or practically instant) solutions, greedy construction remains a better option. Furthermore, while the problems we experiment on have the same computational complexity as real-world optimization problems, additional work may be needed to handle complex side constraints as often seen in industrial problems.
+
+# ACKNOWLEDGMENTS
+
+The computational experiments in this work have been performed using the Bielefeld GPU Cluster.
+
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+Liang Xin, Wen Song, Zhiguang Cao, and Jie Zhang. Step-wise deep learning models for solving routing problems. IEEE Transactions on Industrial Informatics, 17(7):4861–4871, 2021.
+
+Cong Zhang, Wen Song, Zhiguang Cao, Jie Zhang, Puay Siew Tan, and Xu Chi. Learning to dispatch for job shop scheduling via deep reinforcement learning. In Advances in Neural Information Processing Systems, volume 33, pp. 1621–1632, 2020.
+
+# A EXPERIMENTS FOR MORE REALISTIC CVRP INSTANCES
+
+We provide results from additional experiments for the CVRP on more realistic instances to show that our approach is effective at solving instances with a wide range of structures. Our EAS methods are implemented in the same way as in Section 4.2
+
+Setup We evaluate EAS on 9 instance sets from Hottung & Tierney (2020) (consisting of 20 instances each) that have been generated based on the instances from Uchoa et al. (2017) performing 3 runs per instance. The characteristics of the instances vary significantly between sets, but all instances in the same set have been sampled from an identical distribution. For each instance set we train a new model for 3 weeks on a separate, corresponding training set. For testing, we run all (efficient) active search approaches for 200 iterations using the newly trained models. Additionally, we test the generalization performance by solving all instance sets with EAS-Lay using the CVRP model of Section 4.2 that has been trained on the uniform instances (with $n = 1 0 0$ ) from Kool et al. (2019) and call this Lay\*. We only evaluate the generalization performance of EAS-Lay (the best performing EAS approach from Section 4.2) to keep the computational costs low. In all experiments, we use hyperparameters tuned for the uniform CVRP instances. We compare to NLNS, LKH3 and the state-of-the-art unified hybrid genetic search (GS) from Vidal et al. (2014). As is standard in the operations research literature, we round the distances between customers to the nearest integer. Furthermore, we solve instances sequentially and not in batches of different instances. However, to make better use of the available GPU memory, we solve up to 10 copies of the same instance in parallel for the EAS approaches and for POMO with sampling. The best solution found so far is shared between all runs, which has an impact on the imitation learning loss $\mathcal { L } _ { I L }$ for EAS-Emb and EAS-Lay. For EAS-Tab we set $\tilde { Q } = ( 1 - \bar { \beta } ) \cdot Q + \beta \cdot Q ^ { g l o b }$ , where $Q ^ { g l o b }$ is the lookup table for the best solution over all runs and $\beta$ is linearly increased from 0 to 1 over the course of the search.
+
+Results Table 4 shows the gap to the unified hybrid genetic search and the average runtime per instance for all methods. For EAS-Lay we report the performance of the instance set specific models and additionally the generalization performance when using the model trained on uniform CVRP instances (with $n = 1 0 0$ ). The later results are marked with a star. EAS-Emb and EAS-Lay both find better solution than NLNS and LKH3 on 8 out of the 9 instance sets. EAS-Tab outperforms NLNS and LKH3 on all but two instance sets. As a side note, we have found that the original active search (AS) performs surprisingly well, outperforming LKH3 on 3 instance sets, even though it still cannot surpass our newly proposed EAS methods. The version of EAS-Lay (marked with a star) that uses the model trained on uniform instances with $n = 1 0 0$ performs surprisingly well with gaps between $0 . 2 6 \%$ to $4 . 0 8 \%$ to the GS.
+
+Table 4: Results for the CVRP on the instance sets from Hottung & Tierney (2020).   
+
+<table><tr><td></td><td></td><td colspan="4">Gap to GS in %</td><td colspan="2">POMO</td><td colspan="5"> Avg. Runtime in minutes</td></tr><tr><td>Inst.</td><td>n</td><td>POMO Sam. AS</td><td>POMO-EAS Emb Lay Lay* Tab</td><td></td><td>[NLNS LKH|</td><td></td><td></td><td></td><td>POMO-EAS Sam. AS Emb Lay Lay* TabNLNS LKH GS</td><td></td><td></td><td></td></tr><tr><td>XE 1</td><td>100</td><td></td><td></td><td></td><td></td><td>2.12</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>XE</td><td>128</td><td>0.86 0.65 0.76 0.80</td><td>10.23 0.26 0.260.25</td><td>0.61 0.31| 0.26 0.29</td><td>0.32 0.44</td><td>0.9 0.54</td><td>1.3 1.6</td><td>1.3 1.8</td><td>1.4 2.0</td><td>1.5 0.9 2.1</td><td>3.2</td><td>6.2 0.6 2.0</td></tr><tr><td>XE</td><td>180</td><td>0.51 0.20</td><td>0.09 0.09</td><td>0.54 0.13</td><td>0.58 0.16</td><td>1.3 2.5</td><td>2.2</td><td>3.3</td><td>3.7</td><td>24 3.8</td><td>3.2 3.2</td><td>1.2 1.1 1.4</td></tr><tr><td>3.57 XE</td><td>199</td><td></td><td></td><td></td><td></td><td></td><td></td><td>4.2</td><td>4.7</td><td>3.5</td><td>3.2</td><td>3.6 2.4</td></tr><tr><td>XE 9</td><td>213</td><td>1.50 0.88 1.96 1.30</td><td>0.37 0.45</td><td>1.29 0.80 4.08 0.83</td><td>2.03 2.26</td><td>0.72 3.3 1.09</td><td>2.4</td><td></td><td>5.2</td><td>4.9 5.4</td><td>10.2</td><td>1.1 2.4</td></tr><tr><td>XE 11</td><td>236</td><td>1.42 1.22</td><td>0.64 0.71 0.82 0.84</td><td>1.76 0.94</td><td>0.65</td><td>3.7 0.78 4.0</td><td>2.5 2.7</td><td>4.6 5.2</td><td>5.1</td><td>3.9 5.6 4.8</td><td>10.2</td><td>1.1 3.2</td></tr><tr><td>3 XE</td><td></td><td>1.40 0.88</td><td>0.38 0.56</td><td>2.83 0.80</td><td>0.82</td><td>1.55 7.1</td><td>3.5</td><td>8.7</td><td>6</td><td></td><td>10.2</td><td>5.7 3.6</td></tr><tr><td>XE 15</td><td>268</td><td>1.81 2.17</td><td>0.850.96</td><td>2.51 1.27</td><td>1.81</td><td>1.32 7.3</td><td>3.3</td><td>8.8</td><td></td><td>76</td><td>10.3</td><td>5.8 5.5</td></tr><tr><td>XE 17</td><td>297</td><td>1.66 0.97</td><td>0.44 0.65</td><td>2.15 0.92</td><td>1.41</td><td>1.23 8.9</td><td>3.8</td><td>8.8</td><td>6.8</td><td>9.4</td><td>10.3</td><td>2.5 4.2</td></tr></table>
+
+# B ABLATION STUDY: ACTIVE SEARCH LOSS
+
+We evaluate if applying the imitation learning loss component used by EAS-Emb and EAS-Lay to the original active search can significantly improve the performance. To this end, we solve all test instances using active search with and without the imitation learning loss component. Note that the hyperparameters for each approach have been tuned independently on separate validation set instances. Table 5 shows the results. We observe no significant impact of the imitation learning loss $\mathcal { L } _ { \pi }$ on the performance of active search. This means that active search with imitation learning loss is not competitive with EAS-Lay and EAS-Emb across all problems, even when sharing the same loss function.
+
+![](images/41ab96920d4ee9221016be5023bf0d3e37f416c536ccee2088f21bf8bca9448d.jpg)  
+Figure 4: Influence of $\sigma$ on the solution quality for EAS-Tab
+
+# C PARAMETER SWEEP: EAS-TAB INTENSIFICATION
+
+We investigate the impact of the hyperparameter $\sigma$ on EAS-Tab, which controls the degree of exploitation of the search. By setting $\sigma$ to zero (or very small values) we essentially disable the lookup table, thus examining its impact on the search. We again solve all three problems with different values of $\sigma$ on a subset of test instances. We fix $\alpha$ independently based on tuning on a separate set of validation instances. Figure 4 provides the results for adjusting $\sigma$ . For all three problems, $\sigma = 1 0$ provides the best trade-off between exploration and exploitation. Note that low $\sigma$ values (which reduce the impact of the lookup table updates) hurt performance, meaning that the table based adjustments are effective in all cases.

parse/dev/nO5caZwFwYu/nO5caZwFwYu_content_list.json

@@ -0,0 +1,1454 @@
+[
+    {
+        "type": "text",
+        "text": "EFFICIENT ACTIVE SEARCH FOR COMBINATORIAL OPTIMIZATION PROBLEMS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            99,
+            709,
+            147
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "André Hottung Bielefeld University, Germany andre.hottung@uni-bielefeld.de ",
+        "bbox": [
+            184,
+            170,
+            480,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yeong-Dae Kwon Samsung SDS, Korea y.d.kwon@samsung.com ",
+        "bbox": [
+            614,
+            170,
+            813,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Kevin Tierney Bielefeld University, Germany kevin.tierney@uni-bielefeld.de ",
+        "bbox": [
+            183,
+            233,
+            480,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            311,
+            544,
+            327
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, numerous machine learning based methods for combinatorial optimization problems have been proposed that learn to construct solutions in a sequential decision process via reinforcement learning. While these methods can be easily combined with search strategies like sampling and beam search, it is not straightforward to integrate them into a high-level search procedure offering strong search guidance. Bello et al. (2016) propose active search, which adjusts the weights of a (trained) model with respect to a single instance at test time using reinforcement learning. While active search is simple to implement, it is not competitive with state-of-the-art methods because adjusting all model weights for each test instance is very time and memory intensive. Instead of updating all model weights, we propose and evaluate three efficient active search strategies that only update a subset of parameters during the search. The proposed methods offer a simple way to significantly improve the search performance of a given model and outperform state-of-the-art machine learning based methods on combinatorial problems, even surpassing the well-known heuristic solver LKH3 on the capacitated vehicle routing problem. Finally, we show that (efficient) active search enables learned models to effectively solve instances that are much larger than those seen during training. ",
+        "bbox": [
+            233,
+            343,
+            766,
+            579
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            607,
+            336,
+            622
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In recent years, a wide variety of machine learning (ML) based methods for combinatorial optimization problems have been proposed (e.g., Kool et al. (2019); Hottung et al. (2020)) . While early approaches failed to outperform traditional operations research methods, the gap between handcrafted and learned heuristics has been steadily closing. However, the main potential of ML-based methods lies not only in their ability to outperform existing methods, but in automating the design of customized heuristics in situations where no handcrafted heuristics have yet been developed. We hence focus on developing approaches that require as little additional problem-specific knowledge as possible. ",
+        "bbox": [
+            174,
+            638,
+            825,
+            750
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Existing ML based methods for combinatorial optimization problems can be classified into construction methods and improvement methods. Improvement methods search the space of complete solutions by iteratively refining a given start solution. They allow for a guided exploration of the search space and are able to find high-quality solutions. However, they usually rely on problemspecific components. In contrast, construction methods create a solution sequentially starting from an empty solution (i.e., they consider a search space consisting of incomplete solutions). At test time, they can be used to either greedily construct a single solution or to sample multiple solutions from the probability distribution encoded in the trained neural network. Furthermore, the sequential solution generation process can be easily integrated into a beam search without requiring any problem-specific components. However, search methods like sampling and beam search offer no (or very limited) search guidance. Additionally, these methods do not react towards the solutions seen so far, i.e., the underlying distribution from which solutions are sampled is never changed throughout the search. ",
+        "bbox": [
+            174,
+            757,
+            825,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Bello et al. (2016) propose a generic search strategy called active search that allows an extensive, guided search for construction methods without requiring any problem specific components. Active search is an iterative search method that at each iteration samples solutions for a single test instance using a given model and then adjusts the parameters of that model with the objective to increase the likelihood of generating high-quality solutions in future iterations. They report improved performance over random sampling when starting the search from an already trained model. Despite promising results, active search has not seen adaption in the literature. The reason for this is its resource requirements, as adjusting all model parameters separately for each test instance is very time intensive, especially compared to methods that can sample solutions to multiple different instances in one batch. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            229
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We extend the idea of active search as follows. (1) We propose to only adjust a subset of (model) parameters to a single instance during the search, while keeping all other parameters fixed. We show that this efficient active search (EAS) drastically reduces the runtime of active search without impairing the solution quality. (2) We implement and evaluate three different implementations of EAS and show that all offer significantly improved performance over pure sampling approaches. ",
+        "bbox": [
+            174,
+            236,
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+            305
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In our EAS implementations, the majority of (model) parameters are not updated during the search, which drastically reduces the runtime, because gradients only need to be computed for a subset of model weights, and most operations can be applied identically across a batch of different instances. Furthermore, we show that for some problems, EAS finds even better solutions than the original active search. All EAS implementations can be easily applied to existing ML construction methods. ",
+        "bbox": [
+            174,
+            313,
+            825,
+            382
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the proposed EAS approaches on the traveling salesperson problem (TSP), the capacitated vehicle routing problem (CVRP) and the job shop scheduling problem (JSSP). For all problems, we build upon already existing construction approaches that only offer limited search capabilities. In all experiments, EAS leads to significantly improved performance over sampling approaches. For the CVRP and the JSSP, the EAS approaches outperform all state-of-the-art ML based approaches, and even the well-known heuristic solver LKH3 for the CVRP. Furthermore, EAS approaches assists in model generalization, resulting in drastically improved performance when searching for solutions to instances that are much larger than the instances seen during model training. ",
+        "bbox": [
+            174,
+            388,
+            825,
+            501
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 LITERATURE REVIEW ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            535,
+            383,
+            551
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Construction methods Hopfield (1982) first used a neural network (a Hopfield network) to solve small TSP instances with up to 30 cities. The development of recent neural network architectures has paved the way for ML approaches that are able to solve large instances. The pointer network architecture proposed by Vinyals et al. (2015) efficiently learns the conditional probability of a permutation of a given input sequence, e.g., a permutation of cities for a TSP solution. The authors solve TSP instances with up to 50 cities via supervised learning. Bello et al. (2016) report that training a pointer network via actor-critic RL instead results in a better performance on TSP instances with 50 and 100 cities. Furthermore, graph neural networks are used to solve the TSP, e.g., a graph embedding network in Khalil et al. (2017) and a graph attention network in Deudon et al. (2018). ",
+        "bbox": [
+            174,
+            575,
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+            702
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The first applications of neural network based methods to the CVRP are reported by Nazari et al. (2018) and Kool et al. (2019). Nazari et al. (2018) propose a model with an attention mechanism and a recurrent neural network (RNN) decoder that can be trained via actor-critic RL. Kool et al. (2019) propose an attention model that uses an encoder that is similar to the encoder used in the transformer architecture Vaswani et al. (2017). Peng et al. (2019) and Xin et al. (2021) extend the attention model to update the node embeddings throughout the search, resulting in improved performance at the cost of longer runtimes for the CVRP. Falkner & Schmidt-Thieme (2020) propose an attention-based model that constructs tours in parallel for the CVRP with time windows. ",
+        "bbox": [
+            174,
+            708,
+            825,
+            819
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While ML-based construction methods have mainly focused on routing problems, there are some notable exceptions. For example, Khalil et al. (2017) use a graph embedding network approach to solve the minimum vertex cover and the maximum cut problems (in addition to the TSP). Zhang et al. (2020) propose a graph neural network based approach for the job shop scheduling problem (JSSP). Li et al. (2018) use a guided tree search enhanced ML approach to solve the maximal independent set, minimum vertex cover, and the maximal clique problems. For a more detailed review of ML methods on different combinatorial optimization problems, we refer to Vesselinova et al. (2020). ",
+        "bbox": [
+            174,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While most approaches construct routing problem solutions autoregressively, some approaches predict a heat-map that describes which edges will likely be part of a good solution. The heat-map is then used in a post-hoc search to construct solutions. Joshi et al. (2019) use a graph convolutional network to create a heat-map and a beam search to search for solutions. Similarly, Fu et al. (2020) use a graph convolutional residual network with Monte Carlo tree search to solve large TSP instances. Kool et al. (2021) use the model from Joshi et al. (2019) to generate the heat-map and use it to search for solution to TSP and CVRP instances with a dynamic programming based approach. ",
+        "bbox": [
+            173,
+            103,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Improvement methods Improvement methods integrate ML based methods into high-level search heuristics or try to learn improvement operators directly. In general, they often invest more time into solving an instance than construction based methods (and usually find better solutions). Chen & Tian (2019) propose an approach that iteratively changes a local part of the solution. At each iteration, the trainable region picking policy selects a part of the solution that should be changed and a trainable rule picking policy selects an action from a given set of possible modification operations. Hottung & Tierney (2020) propose a method for the CVRP that iteratively destroys parts of a solution using predefined, handcrafted operators and then reconstructs them with a learned repair operator. Wu et al. (2021) and de O. da Costa et al. (2020) propose to use RL to pick an improving solution from a specified local neighborhood (e.g., the 2-Opt neighborhood) to solve routing problems. Hottung et al. (2021) learn a continuous representation of discrete routing problem solutions using conditional variational autoencoders and search for solutions using a generic, continuous optimizer. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 SOLVING COMBINATORIAL OPTIMIZATION PROBLEMS WITH EAS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            404,
+            743,
+            417
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We propose three EAS implementations that adjust a small subset of (model) parameters in an iterative search process. Given an already trained model, we investigate adjusting (1) the normally static embeddings of the problem instance that are generated by the encoder model, (2) the weights of additional instance-specific residual layers added to the decoder, and (3) the parameters of a lookup table that directly affect the probability distribution returned by model. In each iteration, multiple solutions are sampled for one instance and the dynamic (model) parameters are adjusted with the goal of increasing the probability of generating high quality solutions (as during model training). This allows the search to sample solutions of higher quality in subsequent iterations, i.e., the search can focus on the more promising areas of the search space. Once a high-quality solution for an instance is found, the adjusted parameters are discarded, so that the search process can be repeated on other instances. All strategies efficiently generate solutions to a batch of instances in parallel, because the network layers not updated during the search are applied identically to all instances of the batch. ",
+        "bbox": [
+            174,
+            438,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Background RL based approaches for combinatorial problems aim to learn a neural network based model $p _ { \\theta } ( \\pi | l )$ with weights $\\theta$ that can be used to generate a solution $\\pi$ given an instance l. State-ofthe-art approaches usually use a model that consists of an encoder and a decoder unit. The encoder usually creates static embeddings $\\omega$ that describe the instance $l$ using a computationally expensive encoding process (e.g., Kool et al. (2019); Kwon et al. (2020)). The static embeddings are then used to autoregessively construct solutions using the decoder over $T$ time steps. At each step $t$ , the decoder $q _ { \\phi } ( a | s _ { t } , \\bar { \\omega } )$ , with weights $\\phi \\subset \\theta$ , outputs a probability value for each possible action $a$ in the state $s _ { t }$ (e.g., for the TSP, each action corresponds to visiting a different city next). The starting state $s _ { 1 }$ describes the problem instance $l$ (e.g., the positions of the cities for the TSP and the starting city) and the state $s _ { t + 1 }$ is obtained by applying the action $a _ { t }$ selected at time step $t$ to the state $s _ { t }$ . The (partial) solution $\\pi _ { t }$ is defined by the sequence of selected actions $a _ { 1 } , a _ { 2 } , \\ldots , a _ { t }$ . Once a complete solution, $\\pi _ { T }$ , fulfilling all constraints of the problem is constructed, the objective function value $\\mathbf { \\bar { \\boldsymbol { C } } } ( \\pi , l )$ of the solution can be computed (e.g., the tour length for the TSP). ",
+        "bbox": [
+            174,
+            611,
+            825,
+            791
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Figure 1 shows the solution generation for a TSP instance with a model that uses static embeddings. The static embeddings $\\omega$ are used at each decoding step to generate a probability distribution over all possible next actions and the selected action is provided to the decoder in the next decoding step. During testing, solutions can be constructed by either selecting actions greedily or by sampling each action according to $q _ { \\phi } ( a | s _ { t } , \\omega )$ . Since the static embeddings are not updated during solution generation they only need to be computed once per instance, which allows to quickly sample multiple solutions per instance. We note that not all models use static embeddings. Some approaches update all instance embeddings after each action (e.g., Zhang et al. (2020)), which allows the embeddings to contain information on the current solution state $s _ { t }$ . ",
+        "bbox": [
+            174,
+            797,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/3abde8df1e97587e52664ac4987fda3a29e2025873cd251133e478eda9740eb0.jpg",
+        "image_caption": [
+            "Figure 1: Sampling a solution for the TSP with a model $p _ { \\theta } ( \\pi | l )$ that uses static instance embeddings. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            303,
+            98,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 EMBEDDING UPDATES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            237,
+            369,
+            251
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our first proposed strategy, called EAS-Emb, updates the embeddings $\\omega$ generated by an encoder using a loss function consisting of an RL component $\\mathcal { L } _ { R L }$ and an imitation learning $\\left( \\operatorname { I L } \\right)$ component $\\mathcal { L } _ { I L }$ . The loss $\\mathcal { L } _ { R L }$ is based on REINFORCE (Williams, 1992) and is the expected cost of the generated solutions, $\\mathbb { E } \\left[ C ( \\pi ) \\right]$ . We aim to adjust the embedding parameters to increase the likelihood of generating solutions with lower costs (e.g., a shorter tour length for the TSP). The loss $\\mathcal { L } _ { I L }$ is the negation of the log-probability of (re-)generating the best solution seen so far. We adjust the embedding parameters to increase this probability. ",
+        "bbox": [
+            173,
+            263,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "More formally, for an instance $l$ we generate the embeddings $\\omega$ using a given encoder. Based on $\\omega$ we can (repeatedly) sample a solution $\\pi$ whose cost is $C ( \\pi )$ . A subset of the embeddings $\\hat { \\omega } \\subseteq \\omega$ is adjusted to minimize $\\mathcal { L } _ { R L }$ using the gradient ",
+        "bbox": [
+            174,
+            368,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/189abc5d6fb79dfe56b65ebb32b7d500676d329066d509c67e35215326a97182.jpg",
+        "text": "$$\n\\nabla _ { \\hat { \\omega } } \\mathcal { L } _ { R L } ( \\hat { \\omega } ) = \\mathbb { E } _ { \\boldsymbol \\pi } \\left[ ( C ( \\boldsymbol \\pi ) - b _ { \\circ } ) \\nabla _ { \\hat { \\omega } } \\log q _ { \\phi } ( \\boldsymbol \\pi \\mid \\hat { \\omega } ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { q _ { \\phi } ( \\pi \\mid \\hat { \\omega } ) \\equiv \\prod _ { t = 1 } ^ { T } q _ { \\phi } ( a _ { t } \\mid s _ { t } , \\hat { \\omega } ) } \\end{array}$ , and $b _ { \\circ }$ is a baseline (we use the baseline proposed in Kwon et al. (2020) for our experiments). ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For the second loss $\\mathcal { L } _ { I L }$ , let $\\bar { \\pi }$ be the best solution found so far for the instance $l$ , that consists of the actions $\\bar { a } _ { 1 } , \\dots , \\bar { a } _ { T }$ . We use teacher forcing to make the decoder $q _ { \\phi } ( \\cdot | s _ { t } , \\hat { \\omega } )$ generate the solution $\\bar { \\pi }$ , during which we obtain the probability values associated with the actions $\\bar { a } _ { 1 } , \\dots , \\bar { a } _ { T }$ . We increase the log-likelihood of generating $\\bar { \\pi }$ by adjusting $\\hat { \\omega }$ using the gradient ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/02db29bd1a00a2d24674766123c6311de2c249ba381d50637f18cb95eaf93a9c.jpg",
+        "text": "$$\n\\nabla _ { \\boldsymbol { \\hat { \\omega } } } \\mathcal { L } _ { I L } ( \\boldsymbol { \\hat { \\omega } } ) = - \\nabla _ { \\boldsymbol { \\hat { \\omega } } } \\log q _ { \\phi } ( \\bar { \\pi } \\mid \\boldsymbol { \\hat { \\omega } } ) \\equiv - \\nabla _ { \\boldsymbol { \\hat { \\omega } } } \\log \\prod _ { t = 1 } ^ { T } q _ { \\phi } ( \\bar { a } _ { t } | s _ { t } , \\boldsymbol { \\hat { \\omega } } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The gradient of the overall loss $\\mathcal { L } _ { R I L }$ is defined as $\\nabla _ { \\hat { \\omega } } \\mathcal { L } _ { R I L } ( \\hat { \\omega } ) = \\nabla _ { \\hat { \\omega } } \\mathcal { L } _ { R L } ( \\hat { \\omega } ) + \\lambda \\cdot \\nabla _ { \\hat { \\omega } } \\mathcal { L } _ { I L } ( \\hat { \\omega } )$ , where $\\lambda$ is a tunable parameter. If a high value for $\\lambda$ is selected, the search focuses on generating solutions that are similar to the incumbent solution. This accelerates the convergence of the search policy, which is useful when the number of search iterations is limited. ",
+        "bbox": [
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+        "type": "text",
+        "text": "We note that both decoding processes required for RL and $\\mathrm { I L }$ can be carried out in parallel, using the same forward pass through the network. Furthermore, only the parameters $\\hat { \\omega }$ are instance specific, while all other model parameters are identical for all instances. This makes parallelization of multiple instances in a batch more efficient both in time and memory. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.2 ADDED-LAYER UPDATES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We next propose EAS-Lay, which adds an instance-specific residual layer to a trained model. During the search, the weights in the added layer are updated, while the weights of all other original layers are held fixed. We use both RL and $\\mathrm { I L }$ , similarly to EAS-Emb in Section 3.1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We formalize EAS-Lay as follows. For each instance $l$ we insert a layer ",
+        "bbox": [
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+    {
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+        "img_path": "images/a4efe175befc0bb45d9375e239084557e813c70bf10e86d2d14bc5459fc391f8.jpg",
+        "text": "$$\n\\operatorname { L } ^ { \\star } ( h ) = h + ( ( \\operatorname { R e L u } ( h W ^ { 1 } + b ^ { 1 } ) W ^ { 2 } + b ^ { 2 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "into the given decoder $q _ { \\phi }$ , resulting in a slightly modified model $\\tilde { q } _ { \\phi , \\psi }$ , where $\\psi = \\{ W ^ { 1 } , b ^ { 1 } , W ^ { 2 } , b ^ { 2 } \\}$ . The layer takes in the input $h$ and applies two linear transformations with a ReLu activation function in between. The weight matrices $W ^ { \\bar { 1 } }$ and $W ^ { 2 }$ and the bias vectors $b ^ { 1 }$ and $b ^ { 2 }$ are adjusted throughout the search via gradient descent. The weights in the matrix $W ^ { 2 }$ and the vector $b ^ { 2 }$ are initialized to zero so that the added layer does not affect the output of the model during the first iteration of the search. The gradient for $\\mathcal { L } _ { R L }$ is given as ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c1609797b2d752b07426872754f67e6261c30974ae005d79b301a2100da2fcbd.jpg",
+        "text": "$$\n\\nabla _ { \\psi } \\mathcal { L } _ { R L } ( \\psi ) = \\mathbb { E } _ { \\pi } \\big [ ( C ( \\pi ) - b _ { \\circ } ) \\nabla _ { \\psi } \\log \\tilde { q } _ { \\phi , \\psi } ( \\pi ) \\big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            137,
+            660,
+            157
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "with $\\begin{array} { r } { \\tilde { q } _ { \\phi , \\psi } ( \\pi ) \\equiv \\prod _ { t = 1 } ^ { T } \\tilde { q } _ { \\phi , \\psi } ( a _ { t } \\mid s _ { t } , \\omega ) } \\end{array}$ , and $b _ { \\circ }$ is a baseline. The gradient for $\\mathcal { L } _ { I L }$ is defined similarly. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that the majority of the network operations are not instance specific. They can be applied identically to all instances running in parallel as a batch, resulting in significantly lower runtime during search. The position at which the new layer is inserted has an impact on the performance of EAS-Lay, and identifying the best position usually requires testing. In general, the memory requirement of this approach can be reduced by inserting the additional layer closer towards the output layer of the network. This decreases the number of layers to be considered during backpropagation. We noticed for transformer-based architectures that applying the residual layer $\\mathrm { L } ^ { \\star } ( \\cdot )$ to the query vector $q$ before it is passed to the single attention head usually results in a good performance. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.3 TABULAR UPDATES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "EAS-Emb and EAS-Lay require significantly less memory per instance than the original active search. However, they still need to store many gradient weights associated with multiple layers for the purpose of backpropagation. This significantly limits the number of solutions one can generate in parallel. We hence propose EAS-Tab, which does not require backpropagation, but instead uses a simple lookup table to modify the policy of the given model. For each action at a given state, the table provides a guide on how to change its probability, so that the sampled solution has a higher chance at being similar to the best solution found in the past. ",
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+    {
+        "type": "text",
+        "text": "Formally, at each step $t$ during the sequential generation of a solution, we redefine the probability of selecting action $a _ { t }$ in the state $s _ { t }$ as $q _ { \\phi } ( a | \\bar { s _ { t } } , \\omega ) ^ { \\alpha } \\cdot Q _ { g ( s _ { t } , a _ { t } ) }$ and renormalize over all possible actions using the softmax function. Here, $\\alpha$ is a hyperparameter, and $g$ is a function that maps each possible state and action pair to an entry in the table $Q$ . The network parameters $\\theta$ remain unchanged, resulting in fast and memory efficient solution generation. The hyperparameter $\\alpha$ is similar to the temperature value proposed in Bello et al. (2016) and modifies the steepness of the probability distribution returned by the model (lower values increase the exploration of the search). During search, the table $Q$ is updated with the objective of increasing the quality of the generated solutions. More precisely, after each iteration, $Q$ is updated based on the best solution $\\bar { \\pi }$ found so far consisting of the actions $\\bar { a } _ { 1 } , \\dots , \\bar { a } _ { T }$ at states $\\bar { s } _ { 1 } , \\dots , \\bar { s } _ { T }$ , respectively, with ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/dafda474a5834db9ed417690205a5bb2e7098e1162c71613804c95c313b3d37d.jpg",
+        "text": "$$\nQ _ { g ( s _ { t } , a _ { t } ) } = \\left\\{ \\begin{array} { l l } { \\operatorname* { m a x } ( 1 , \\frac { \\sigma } { q _ { \\phi } ( a | s _ { t } , \\omega ) ^ { \\alpha } } ) , } & { \\mathrm { i f } g ( s _ { t } , a _ { t } ) \\in \\{ g ( \\bar { a } _ { 1 } , \\bar { s } _ { 1 } ) , \\dots , g ( \\bar { a } _ { T } , \\bar { s } _ { T } ) \\} } \\\\ { 1 , } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The hyperparameter $\\sigma$ defines the degree of exploitation of the search. If a higher value of $\\sigma$ is used, the probabilities for actions that generate the incumbent solution are increased. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In contrast to embedding or added-layer updates, this EAS method requires deeper understanding of the addressed combinatorial optimization problem to design the function $g ( s _ { t } , a _ { t } )$ . For example, for the TSP with $n$ nodes we use a table $Q$ of size $n \\times n$ in which each entry $Q _ { i , j }$ corresponds to a directed edge $e _ { i , j }$ of the problem instance. The probability increases for the same directed edge that was used in the incumbent solution. This definition of $g ( s _ { t } , a _ { t } )$ effectively ignores the information on all the previous visits stored in state $s _ { t }$ , focusing instead on the current location (city) in choosing the next move. We note that this EAS approach is similar to the ant colony optimization algorithm (Dorigo et al., 2006), which has been applied to a wide variety of combinatorial optimization problems. ",
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+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We evaluate all EAS strategies using existing, state-of-the-art RL based methods for three different combinatorial optimization problems. For the first two, the TSP and the CVRP, we implement EAS for the POMO approach (Kwon et al., 2020). For the third problem, the JSSP, we use the L2D method from Zhang et al. (2020). We extend the code made available by the authors of POMO (MIT license) and L2D (no license) with our EAS strategies to ensure a fair evaluation. Note that we only make minor modifications to these methods, and we use the models trained by the authors when available. ",
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+        "type": "text",
+        "text": "We run all experiments on a GPU cluster using a single Nvidia Tesla V100 GPU and a single core of an Intel Xeon 4114 CPU at $2 . 2 \\ : \\mathrm { G H z }$ for each experiment. Our source code is available at https://github.com/ahottung/EAS. We use the Adam optimizer (Kingma & Ba, 2014) for all EAS approaches. The hyperparameters $\\lambda , \\sigma , \\alpha$ , and the learning rate for the optimizer are tuned via Bayesian optimization using scikit-optimize (Head et al., 2020) on separate validation instances, which are sampled from the same distribution as the test instances. The hyperparameters are not adjusted for larger instances used to evaluate the generalization performance. ",
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+    {
+        "type": "text",
+        "text": "4.1 TSP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The TSP is a well-known routing problem involving finding the shortest tour between a set of $n$ nodes (i.e., cities) that visits each node exactly once and returns to the starting node. We assume that the distance matrix obeys the triangle inequality. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Implementation POMO uses a model that is very similar to the AM model from Kool et al. (2019). The model generates instance embeddings only once per instance and does not update them during construction. The probability distribution over all actions are generated by a decoder, whose last layer is a single-headed attention layer. This last layer calculates the compatibility of a query vector $q$ to the key vector $k _ { i }$ for each node $i$ . In this operation, the key vector $k _ { i }$ is an embedding that has been computed separately, but identically for each input (i.e., node $i$ ) during the instance encoding process. For EAS-Emb, we only update the set of single-head keys $k _ { i }$ $( i = 1 , \\ldots , n )$ . For EAS-Lay we apply the residual layer $\\mathrm { L } ^ { \\star } ( \\cdot )$ described in Equation 3 to the query vector $q$ before it is passed to the single attention head. For EAS-Tab, we use a table $Q$ of size $n \\times n$ and the mapping function $g ( s _ { t } , a _ { t } )$ such that each entry $Q _ { i , j }$ corespondents to a directed edge $e _ { i , j }$ of the problem instance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Setup We use the 10,000 TSP instances with $n = 1 0 0$ from Kool et al. (2019) for testing and three additional sets of 1,000 instances to evaluate generalization performance. We evaluate the EAS approaches against just using POMO with greedy action selection, random sampling, and active search as in Bello et al. (2016). In all cases, we use the model trained on instances with $n = 1 0 0$ made available by the POMO authors. For greedy action selection, POMO generates $8 \\cdot n$ solutions for an instance of size $n$ (using 8 augmentations and $n$ different starting cities). In all other cases, we generate $2 0 0 \\cdot 8 \\cdot n$ solutions per instance (over the course of 200 iterations for the (E)AS approaches). The batch size (the number of instances solved in parallel) is selected for each method individually to fully utilize the available GPU memory. We compare to the exact solver Concorde (Applegate et al., 2006), the heuristic solver LKH3 (Helsgaun, 2017), the graph convolutional neural network with beam search (GCN-BS) from Joshi et al. (2019), the 2-Opt based deep learning (2-Opt-DL) approach from de O. da Costa et al. (2020), the learning improvement heuristics (LIH) method from Wu et al. (2021), the conditional variational autoencoder (CVAE-Opt) approach (Hottung et al., 2021), and deep policy dynamic programming (DPDP) (Kool et al., 2021). ",
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+    {
+        "type": "text",
+        "text": "Results Table 1 shows the average costs, average gap and the total runtime (wall-clock time) for each instance set. The exact solver Concorde performs best overall, as it is a highly specialized TSP solver. Of the POMO-based approaches, the original active search offers the best gap to optimality, but requires 5 days of runtime. EAS significantly lowers the runtime while the gap is only marginally larger. DPDP performs best among ML-based approaches. However, DPDP relies on a handcrafted and problem-specific beam search, whereas EAS methods are completely problem-independent. On the larger instances, EAS significantly improves generalization performance, reducing the gap over sampling by up to $3 . 6 \\mathrm { x }$ . We also evaluate active search using the imitation learning loss, but observe no impact on the search performance (see Appendix B). ",
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+    {
+        "type": "table",
+        "img_path": "images/b850778457b82bd499ebcca9bc2866dca78cef43add4b0a344aaaf470b378364.jpg",
+        "table_caption": [
+            "Table 1: Results for the TSP "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">Testing (10k inst.)</td><td colspan=\"10\">Generalization (1k instances)</td></tr><tr><td colspan=\"3\">n=100</td><td colspan=\"3\">n=125</td><td colspan=\"3\">n=150</td><td colspan=\"3\"></td><td>n = 200</td></tr><tr><td>Method</td><td>Obj.</td><td>Gap</td><td>Time</td><td>Obj.</td><td>Gap</td><td>Time</td><td>Obj.</td><td>Gap</td><td></td><td>Time</td><td>Obj.</td><td>Gap</td><td>Time</td></tr><tr><td>Concorde</td><td></td><td>[7.765 0.000%</td><td></td><td></td><td>82M|8.583 0.000%</td><td></td><td></td><td>12M|9.346 0.000%</td><td></td><td></td><td>17M|10.687 0.000%</td><td></td><td>31M</td></tr><tr><td>LKH3</td><td>7.765</td><td>0.000%</td><td>8H</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>8.5830.000%73M9.3460.000% 99M10.6870.000%</td><td>3H</td></tr><tr><td>GCN-BS</td><td>7.87</td><td>1.39%</td><td>40M</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td><td>-</td></tr><tr><td>2-Opt-DL</td><td>7.83</td><td>0.87%</td><td>41M</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td></tr><tr><td>LIH</td><td>7.87</td><td>1.42%</td><td>2H</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td></tr><tr><td>CVAE-Opt</td><td>1</td><td>0.343%</td><td>6D</td><td></td><td>8.646 0.736%</td><td>21H</td><td></td><td>9.482 1.454%</td><td></td><td>30H</td><td></td><td></td><td></td></tr><tr><td>DPDP</td><td>7.765</td><td>0.004%</td><td>2H</td><td></td><td>8.589 0.070%</td><td>31M</td><td></td><td>9.4340.942%</td><td></td><td>44M</td><td></td><td>11.154 4.370%</td><td>74M</td></tr><tr><td>Greedy</td><td>7.776</td><td>0.146%</td><td>1M|</td><td>8.607</td><td>0.278%</td><td></td><td>&lt;1M|9.397</td><td></td><td>0.542%</td><td>&lt;1M|</td><td>10.843</td><td>1.457%</td><td>1M</td></tr><tr><td>Sampling</td><td>7.770</td><td>0.074%</td><td>4H</td><td>8.595</td><td>0.145%</td><td>45M</td><td></td><td>9.378</td><td>0.334%</td><td>78M</td><td></td><td>10.8381.416%</td><td>3H</td></tr><tr><td>Active S.</td><td></td><td>7.768 0.046%</td><td>5D</td><td>8.591</td><td>0.095%</td><td>15H</td><td>9.364</td><td></td><td>0.192%</td><td>19H</td><td>10.735</td><td>50.447%</td><td>24H</td></tr><tr><td>0 EAS-Emb</td><td></td><td>7.769 0.063%</td><td>5H</td><td>8.591</td><td>0.092%</td><td>57M</td><td></td><td>9.363</td><td>0.174%</td><td>2H</td><td>10.730 0.400%</td><td></td><td>4H</td></tr><tr><td>P EAS-Lay</td><td></td><td>7.769 0.053%</td><td>7H</td><td>8.591</td><td>0.089%</td><td>74M</td><td></td><td>9.363 0.176%</td><td></td><td>2H</td><td>10.737 0.471%</td><td></td><td>4H</td></tr><tr><td>EAS-Tab</td><td></td><td>7.768 0.048%</td><td>5H</td><td>8.591</td><td>0.091%49M</td><td></td><td></td><td>9.3650.196%</td><td></td><td>1H</td><td>10.756 0.650%</td><td></td><td>3H</td></tr></table>",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "4.2 CVRP ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The goal of the CVRP is to find the shortest routes for a set of vehicles with limited capacity that must deliver goods to a set of $n$ customers. We again use the POMO approach as a basis for our EAS strategies. As is standard in the ML literature, we evaluate all approaches on instance sets where the locations and demands are sampled uniformly at random. Additionally, we consider the more realistic instance sets proposed in Hottung & Tierney (2020) with up to 297 customers (see Appendix A). ",
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+    {
+        "type": "text",
+        "text": "Implementation We use the same EAS implementation for the CVRP as for the TSP. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Setup We use the 10,000 CVRP instances from Kool et al. (2019) for testing and additional sets of 1,000 instances to evaluate the generalization performance. Again, we compare the EAS approaches to POMO using greedy action selection, sampling and active search. We generate the same number of solutions per instance as for the TSP. We compare to LIH, CAVE-Opt, DPDP, NeuRewriter (Chen & Tian, 2019) and neural large neighborhood search (NLNS) from Hottung & Tierney (2020). ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Results Table 2 shows the average costs, the average gap to LKH3 and the total wall-clock time for all instance sets. EAS-Lay outperforms all other approaches on the test instances, including approaches that rely on problem-specific knowledge, with a gap that beats LKH3. Both other EAS methods also find solutions of better quality than LKH3, which is quite an accomplishment given the many years of work on the LKH3 approach. We note it is difficult to provide a fair comparison between a single-core, CPU-bound technique like LKH3 and our approaches that use a GPU. Nonetheless, assuming a linear speedup, at least 18 CPU cores would be needed for LKH3 to match the runtime of EAS-Tab. On the generalization instance sets with $n = 1 2 5$ and $n = 1 5 0$ , the EAS approaches also outperform LKH3 and CVAE-Opt while being significantly faster than active search. On the instances with $n = 2 0 0$ , active search finds the best solutions of all POMO based approaches with a gap of $0 . 2 2 \\%$ to LKH3, albeit with a long runtime of 36 hours. We hypothesize that significant changes to the learned policy are necessary to generate high-quality solutions for instances that are very different to those seen during training. Active search’s ability to modify all model parameters makes it easier to make those changes. EAS-Tab offers the worst performance on the instances with $n = 2 0 0$ with a gap of $1 1 . 8 \\%$ . This is because EAS-Tab is very sensitive to the selection of the hyperparameter $\\alpha$ , meaning that EAS-Tab requires hyperparameter tuning on some problems to generalize more effectively. Adjusting $\\alpha$ for the $n = 2 0 0$ case improves EAS-Tab’s gap to at least $3 . 5 4 \\%$ , making it slightly better than greedy or sampling. ",
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/acaba43ae1df91044310bfda809b60ffddcc8cd8cff8313d303fd2d71b71174d.jpg",
+        "table_caption": [
+            "Table 2: Results for the CVRP on instances with uniformly sampled locations and demands "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">Testing (10k inst.) n=100</td><td colspan=\"10\">Generalization (1k instances)</td></tr><tr><td>Obj.</td><td>Gap</td><td>Time</td><td>Obj.</td><td>n =125 Gap</td><td>Time</td><td>Obj.</td><td>n=150 Gap</td><td>Time</td><td>Obj.</td><td>n = 200 Gap</td><td></td><td>Time</td></tr><tr><td>Method</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>LKH3</td><td>|15.65</td><td>0.00%</td><td></td><td>6D|17.50</td><td>0.00%</td><td></td><td>19H|19.22</td><td></td><td>0.00%</td><td></td><td>20H|22.00</td><td>0.00%</td><td>25H</td></tr><tr><td>NLNS NeuRewriter</td><td>15.99</td><td>2.23%</td><td>62M</td><td>[|18.07</td><td>3.23%</td><td></td><td>9M|19.96</td><td></td><td>3.86%</td><td>12M|23.02</td><td></td><td>4.66%</td><td>24M</td></tr><tr><td>LIH</td><td>16.10</td><td></td><td>66M</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td></tr><tr><td>CVAE-Opt</td><td>16.03</td><td>2.47% 1.36%</td><td>5H</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td><td>=</td></tr><tr><td>DPDP</td><td>15.63</td><td>-0.13%</td><td>11D 23H</td><td>17.87 17.51</td><td>2.08% 0.07%</td><td>36H 3H</td><td>19.84 19.31</td><td></td><td>3.24% 0.48%</td><td>46H 5H</td><td>22.26</td><td>51.20%</td><td>= 9H</td></tr><tr><td>Greedy</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>15.76</td><td>0.76%</td><td>2M</td><td>17.73</td><td>1.29%</td><td>&lt;1M</td><td>19.64</td><td></td><td>2.18%</td><td>1M</td><td>|22.90</td><td>4.12%</td><td>1M</td></tr><tr><td>Sampling 0</td><td>15.67</td><td>0.17%</td><td>7H</td><td>17.60</td><td>0.54%</td><td>73M</td><td>19.48</td><td></td><td>1.35%</td><td>2H</td><td>23.18</td><td>5.35%</td><td>5H</td></tr><tr><td>Active S.</td><td>15.63</td><td>-0.07%</td><td>8D</td><td>17.47</td><td>-0.21%</td><td>25H</td><td>19.21</td><td></td><td>-0.03%</td><td>29H</td><td>22.05</td><td>0.22%</td><td>36H</td></tr><tr><td>EAS-Emb</td><td>15.63 15.61</td><td>-0.08%</td><td>9H 12H</td><td>17.47</td><td>-0.21%</td><td>93M</td><td>19.22</td><td></td><td>0.03%</td><td>3H</td><td>22.19</td><td>0.88%</td><td>6H</td></tr><tr><td>EAS-Lay EAS-Tab</td><td>15.62</td><td>-0.23% -0.14%</td><td>8H</td><td>17.50</td><td>17.46 -0.24% 0.00%</td><td>2H 80M</td><td>19.21 19.36</td><td></td><td>-0.04% 0.72%</td><td>3H 2H</td><td>22.10 24.56</td><td>0.45% 11.8%</td><td>8H 5H</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "4.3 JSSP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The JSSP is a scheduling problem involving assigning jobs to a set of heterogeneous machines. Each job consists of multiple operations that are run sequentially on the set of machines. The objective is to minimize the time needed to complete all jobs, called the makespan. We evaluate EAS using the L2D approach, which is a state-of-the-art ML based construction method using a graph neural network. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Implementation L2D represents JSSP instances as disjunctive graphs in which each operation of an instance is represented by a node in the graph. To create a schedule, L2D sequentially selects the operation that should be scheduled next. To this end, an embedding $h _ { v }$ is created for each node $v$ in a step-wise encoding process. In contrast to POMO, the embeddings $h _ { v }$ are recomputed after each decision step $t$ . Since EAS-Emb requires static embeddings, we modify the network to use $\\tilde { h } _ { v } ^ { t } = h _ { v } ^ { t } + h _ { v } ^ { S T }$ as an embedding for node t to zero. During the searc $v$ at step  with E $t$ , where S-Emb $h _ { v } ^ { S T }$ is a vector that is initialized winly adjust the static component $h _ { v } ^ { S T }$ of the embedding with gradient descent. For EAS-Lay, we insert the residual layer $\\mathrm { L } ^ { \\star } ( \\cdot )$ described in Equation 3 to each embedding $h _ { v }$ separately and identically. Finally, for EAS-Tab, we use a table $Q$ of size $| O | \\times | O |$ , where $| O |$ is the number of operations, and we design the function $g ( s _ { t } , a _ { t } )$ so that the entry $Q _ { i , j }$ corresponds to selecting the operation $o _ { j }$ directly after the operation $o _ { i }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Setup We use three instance sets with 100 instances from Zhang et al. (2020) for testing and to evaluate the generalization performance. We use the exact solver Google OR-Tools (Perron & Furnon) as a baseline, allowing it a maximum runtime of 1 hour per instance. Furthermore, we compare to L2D with greedy action selection. Note that the performance of the L2D implementation is CPU bound and does not allow different instances to be batch processed. We hence solve instances sequentially and generate significantly fewer solutions per instance than for the TSP and the CVRP. For sampling, active search and the EAS approaches we sample 8,000 solutions per problem instance over the course of 200 iterations for the (efficient) active search approaches. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Results Table 3 shows the average gap to the OR-Tools solution and the total wall-clock time per instance set. EAS-Emb offers the best performance for all three instance sets. On the $1 0 \\times 1 0$ instances, EAS-Emb reduces the gap by $50 \\%$ in comparison to pure sampling. Even on the $2 0 \\times 1 5$ instances it reduces the gap to $1 6 . 8 \\%$ from $2 0 . 8 \\%$ for pure sampling, despite the low number of sampled solutions per instance. EAS-Lay offers performance that is comparable to active search. We note that if L2D were to more heavily use the GPU, instances could be solved in batches, thus drastically reducing the runtime of EAS-Lay and EAS-Tab. While EAS-Tab shows similar performance to active search on the test instance set, it is unable to generalize effectively to the larger instances. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.4 SEARCH TRAJECTORY ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To get a better understanding of how efficient active search improves performance, we monitor the quality of the solutions sampled at each of the 200 iterations of the search. Figure 2 reports the average quality over all test instances for the JSSP and over the first 1,000 test instances for the TSP and CVRP. As expected, the quality of solutions generated via pure sampling does not change over the course of the search for all three problems. For all other methods, the quality of the generated solutions improves throughout the search. Thus, all active search variants successfully modify the (model) parameters in a way that increases the likelihood of generating high-quality solutions. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f2ce96d91caec68f4b05fe8c69adad675170d3045007f55110ea4d8ab189f66e.jpg",
+        "table_caption": [
+            "Table 3: Results for the JSSP "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\" rowspan=\"2\"></td><td colspan=\"3\">Testing(100 inst.）</td><td colspan=\"4\">Generalization (1OO instances)</td></tr><tr><td colspan=\"2\">10×10</td><td colspan=\"3\">15×15</td><td colspan=\"3\">20×15</td></tr><tr><td colspan=\"2\" rowspan=\"2\">Method</td><td rowspan=\"2\">Obj. Gap</td><td rowspan=\"2\">Time</td><td rowspan=\"2\">Obj.</td><td>Gap</td><td>Time</td><td>Obj. Gap</td><td>Time</td></tr><tr><td rowspan=\"2\"></td><td rowspan=\"2\"></td><td rowspan=\"2\"></td><td rowspan=\"2\">0.0%</td></tr><tr><td rowspan=\"2\">OR-Tools Greedy</td><td rowspan=\"2\">1807.6 0.0%</td></tr><tr><td>37S|1</td><td rowspan=\"2\">[1188.0</td><td colspan=\"2\" rowspan=\"2\">0.0%</td><td rowspan=\"2\">3H| |1345.5</td><td rowspan=\"2\"></td><td rowspan=\"2\">80H</td></tr><tr><td rowspan=\"2\"></td><td rowspan=\"2\">1988.6</td></tr><tr><td>22.3% 871.7</td><td>20S 1528.3</td><td>28.6%</td><td>44S</td><td>1738.0</td><td>29.2%</td><td>60S</td></tr><tr><td rowspan=\"4\">Sampling L</td><td>854.2</td><td>8.0% 5.8%</td><td>8H 8H</td><td>1378.3 16.0% 1345.2 13.2%</td><td>25H 32H</td><td>1624.6 1576.5</td><td>20.8% 17.2%</td><td>40H</td></tr><tr><td>EAS-Emb</td><td>837.0 3.7%</td><td>7H</td><td>1326.4 11.7%</td><td>22H</td><td>1570.8</td><td>16.8%</td><td>50H 37H</td></tr><tr><td>EAS-Lay</td><td>859.6 6.5%</td><td>7H</td><td>1352.6</td><td>13.8%</td><td>25H 1581.8</td><td>17.6%</td><td>46H</td></tr><tr><td>EAS-Tab</td><td>860.2 6.5%</td><td>8H</td><td>1376.8</td><td>15.9% 29H</td><td></td><td>1623.420.7%</td><td>51H</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/aeff2be7dc644f08b6349a6f6cabfb88e1ed829fcc964cb99457f18da9f01b79.jpg",
+        "image_caption": [
+            "Figure 2: Average costs of sampled solutions at each iteration (best viewed in color). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/3814f29ab52dc4112005668d02a29044fe9d34e4442175ae8284d12c9a420a11.jpg",
+        "image_caption": [
+            "Figure 3: Influence of $\\lambda$ on the solution quality for EAS-Emb and EAS-Lay. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For the TSP, EAS-Emb and EAS-Lay offer nearly identical performance, with EAS-Tab outperforming both by a very slight margin. The original active search is significantly more unstable, which is likely the result of the learning rate being too high. Note that the learning rate has been tuned on an independent validation set. These results indicate that selecting a suitable learning rate is significantly more difficult for the original active search than for our efficient active search variants where only a subset of (model) parameters are changed. For the CVRP, all EAS variants find better solutions on average than the original search after only a few iterations. Keeping most parameters fixed seems to simplify the underlying learning problem and allows for faster convergence. For the JSSP, EAS-Emb offers significantly better performance than all other methods. The reason for this is that the L2D approach uses only two node features and has a complex node embedding generation procedure. While the original active search must fine tune the entire embedding generation process to modify the generated solutions, EAS-Emb can just modify the node embedding directly. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.5 ABLATION STUDY: IMITATION LEARNING LOSS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We evaluate the impact of the imitation learning loss $\\mathcal { L } _ { I L }$ of EAS-Emb and EAS-Lay with a sensitivity and ablation analysis for the hyperparameter $\\lambda$ . We solve the first 500 test instances (to reduce the computational costs) for the TSP and CVRP, and all test instances for the JSSP using EAS-Emb and EAS-Lay with different $\\lambda$ values. The learning rate remains fixed to a value determined in independent tuning runs in which $\\lambda$ is fixed to zero. Figure 3 shows the results for all three problems. For the TSP and the CVRP, the results show that $\\mathcal { L } _ { I L }$ can significantly improve performance. When $\\lambda$ is set to 0 or very small values, $\\mathcal { L } _ { I L }$ is disabled, thus including $\\mathcal { L } _ { I L }$ is clearly beneficial on the TSP and CVRP. For the JSSP, the inclusion of $\\mathcal { L } _ { I L }$ does not greatly improve performance, but it does not hurt it, either. Naturally, $\\lambda$ should not be selected too low or too high as either too little or too much intensification can hurt search performance. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We presented a simple technique that can be used to extend ML-based construction heuristics by an extensive search. Our proposed modification of active search fine tunes a small subset of (model) parameters to a single instance at test time. We evaluate three example implementations of EAS that all result in significantly improved model performance in both testing and generalization experiments on three different, difficult combinatorial optimization problems. Our approach of course comes with some key limitations. Search requires time, thus for applications needing extremely fast (or practically instant) solutions, greedy construction remains a better option. Furthermore, while the problems we experiment on have the same computational complexity as real-world optimization problems, additional work may be needed to handle complex side constraints as often seen in industrial problems. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The computational experiments in this work have been performed using the Bielefeld GPU Cluster. ",
+        "bbox": [
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+            127,
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+            143
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A EXPERIMENTS FOR MORE REALISTIC CVRP INSTANCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            673,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We provide results from additional experiments for the CVRP on more realistic instances to show that our approach is effective at solving instances with a wide range of structures. Our EAS methods are implemented in the same way as in Section 4.2 ",
+        "bbox": [
+            174,
+            136,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Setup We evaluate EAS on 9 instance sets from Hottung & Tierney (2020) (consisting of 20 instances each) that have been generated based on the instances from Uchoa et al. (2017) performing 3 runs per instance. The characteristics of the instances vary significantly between sets, but all instances in the same set have been sampled from an identical distribution. For each instance set we train a new model for 3 weeks on a separate, corresponding training set. For testing, we run all (efficient) active search approaches for 200 iterations using the newly trained models. Additionally, we test the generalization performance by solving all instance sets with EAS-Lay using the CVRP model of Section 4.2 that has been trained on the uniform instances (with $n = 1 0 0$ ) from Kool et al. (2019) and call this Lay\\*. We only evaluate the generalization performance of EAS-Lay (the best performing EAS approach from Section 4.2) to keep the computational costs low. In all experiments, we use hyperparameters tuned for the uniform CVRP instances. We compare to NLNS, LKH3 and the state-of-the-art unified hybrid genetic search (GS) from Vidal et al. (2014). As is standard in the operations research literature, we round the distances between customers to the nearest integer. Furthermore, we solve instances sequentially and not in batches of different instances. However, to make better use of the available GPU memory, we solve up to 10 copies of the same instance in parallel for the EAS approaches and for POMO with sampling. The best solution found so far is shared between all runs, which has an impact on the imitation learning loss $\\mathcal { L } _ { I L }$ for EAS-Emb and EAS-Lay. For EAS-Tab we set $\\tilde { Q } = ( 1 - \\bar { \\beta } ) \\cdot Q + \\beta \\cdot Q ^ { g l o b }$ , where $Q ^ { g l o b }$ is the lookup table for the best solution over all runs and $\\beta$ is linearly increased from 0 to 1 over the course of the search. ",
+        "bbox": [
+            173,
+            184,
+            825,
+            450
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Results Table 4 shows the gap to the unified hybrid genetic search and the average runtime per instance for all methods. For EAS-Lay we report the performance of the instance set specific models and additionally the generalization performance when using the model trained on uniform CVRP instances (with $n = 1 0 0$ ). The later results are marked with a star. EAS-Emb and EAS-Lay both find better solution than NLNS and LKH3 on 8 out of the 9 instance sets. EAS-Tab outperforms NLNS and LKH3 on all but two instance sets. As a side note, we have found that the original active search (AS) performs surprisingly well, outperforming LKH3 on 3 instance sets, even though it still cannot surpass our newly proposed EAS methods. The version of EAS-Lay (marked with a star) that uses the model trained on uniform instances with $n = 1 0 0$ performs surprisingly well with gaps between $0 . 2 6 \\%$ to $4 . 0 8 \\%$ to the GS. ",
+        "bbox": [
+            174,
+            457,
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/2b70c77680f3caf606ae0f0a6c0624de6f9421152730ee2f175b081fef75c62a.jpg",
+        "table_caption": [
+            "Table 4: Results for the CVRP on the instance sets from Hottung & Tierney (2020). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td colspan=\"4\">Gap to GS in %</td><td colspan=\"2\">POMO</td><td colspan=\"5\"> Avg. Runtime in minutes</td></tr><tr><td>Inst.</td><td>n</td><td>POMO Sam. AS</td><td>POMO-EAS Emb Lay Lay* Tab</td><td></td><td>[NLNS LKH|</td><td></td><td></td><td></td><td>POMO-EAS Sam. AS Emb Lay Lay* TabNLNS LKH GS</td><td></td><td></td><td></td></tr><tr><td>XE 1</td><td>100</td><td></td><td></td><td></td><td></td><td>2.12</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>XE</td><td>128</td><td>0.86 0.65 0.76 0.80</td><td>10.23 0.26 0.260.25</td><td>0.61 0.31| 0.26 0.29</td><td>0.32 0.44</td><td>0.9 0.54</td><td>1.3 1.6</td><td>1.3 1.8</td><td>1.4 2.0</td><td>1.5 0.9 2.1</td><td>3.2</td><td>6.2 0.6 2.0</td></tr><tr><td>XE</td><td>180</td><td>0.51 0.20</td><td>0.09 0.09</td><td>0.54 0.13</td><td>0.58 0.16</td><td>1.3 2.5</td><td>2.2</td><td>3.3</td><td>3.7</td><td>24 3.8</td><td>3.2 3.2</td><td>1.2 1.1 1.4</td></tr><tr><td>3.57 XE</td><td>199</td><td></td><td></td><td></td><td></td><td></td><td></td><td>4.2</td><td>4.7</td><td>3.5</td><td>3.2</td><td>3.6 2.4</td></tr><tr><td>XE 9</td><td>213</td><td>1.50 0.88 1.96 1.30</td><td>0.37 0.45</td><td>1.29 0.80 4.08 0.83</td><td>2.03 2.26</td><td>0.72 3.3 1.09</td><td>2.4</td><td></td><td>5.2</td><td>4.9 5.4</td><td>10.2</td><td>1.1 2.4</td></tr><tr><td>XE 11</td><td>236</td><td>1.42 1.22</td><td>0.64 0.71 0.82 0.84</td><td>1.76 0.94</td><td>0.65</td><td>3.7 0.78 4.0</td><td>2.5 2.7</td><td>4.6 5.2</td><td>5.1</td><td>3.9 5.6 4.8</td><td>10.2</td><td>1.1 3.2</td></tr><tr><td>3 XE</td><td></td><td>1.40 0.88</td><td>0.38 0.56</td><td>2.83 0.80</td><td>0.82</td><td>1.55 7.1</td><td>3.5</td><td>8.7</td><td>6</td><td></td><td>10.2</td><td>5.7 3.6</td></tr><tr><td>XE 15</td><td>268</td><td>1.81 2.17</td><td>0.850.96</td><td>2.51 1.27</td><td>1.81</td><td>1.32 7.3</td><td>3.3</td><td>8.8</td><td></td><td>76</td><td>10.3</td><td>5.8 5.5</td></tr><tr><td>XE 17</td><td>297</td><td>1.66 0.97</td><td>0.44 0.65</td><td>2.15 0.92</td><td>1.41</td><td>1.23 8.9</td><td>3.8</td><td>8.8</td><td>6.8</td><td>9.4</td><td>10.3</td><td>2.5 4.2</td></tr></table>",
+        "bbox": [
+            183,
+            636,
+            815,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B ABLATION STUDY: ACTIVE SEARCH LOSS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            806,
+            558,
+            821
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We evaluate if applying the imitation learning loss component used by EAS-Emb and EAS-Lay to the original active search can significantly improve the performance. To this end, we solve all test instances using active search with and without the imitation learning loss component. Note that the hyperparameters for each approach have been tuned independently on separate validation set instances. Table 5 shows the results. We observe no significant impact of the imitation learning loss $\\mathcal { L } _ { \\pi }$ on the performance of active search. This means that active search with imitation learning loss is not competitive with EAS-Lay and EAS-Emb across all problems, even when sharing the same loss function. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            922
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/41ab96920d4ee9221016be5023bf0d3e37f416c536ccee2088f21bf8bca9448d.jpg",
+        "image_caption": [
+            "Figure 4: Influence of $\\sigma$ on the solution quality for EAS-Tab "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            125,
+            823,
+            295
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            347,
+            825,
+            376
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C PARAMETER SWEEP: EAS-TAB INTENSIFICATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            396,
+            622,
+            412
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We investigate the impact of the hyperparameter $\\sigma$ on EAS-Tab, which controls the degree of exploitation of the search. By setting $\\sigma$ to zero (or very small values) we essentially disable the lookup table, thus examining its impact on the search. We again solve all three problems with different values of $\\sigma$ on a subset of test instances. We fix $\\alpha$ independently based on tuning on a separate set of validation instances. Figure 4 provides the results for adjusting $\\sigma$ . For all three problems, $\\sigma = 1 0$ provides the best trade-off between exploration and exploitation. Note that low $\\sigma$ values (which reduce the impact of the lookup table updates) hurt performance, meaning that the table based adjustments are effective in all cases. ",
+        "bbox": [
+            173,
+            428,
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+            540
+        ],
+        "page_idx": 12
+    }
+]

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