Deterministic Edge Connectivity and Max Flow using Subquadratic Cut | Queries

Question: Develop a deterministic algorithm to find the global min-cut of a simple graph using sub-quadratic queries in the cut query model, specifically aiming for \( \tilde{O}(n^{5/3}) \) queries. Design an algorithm to find \( s \)-\( t \) max-flows of size \( \tilde{O}(n) \) using \( \tilde{O}(n^{5/3}) \) queries. Additionally, provide efficient cut-query implementations for expander decomposition and isolating cuts.

1. Problem and model

We study deterministic algorithms in the cut-query model for a simple undirected graph G=(V,E), |V|=n. A cut query on S\subseteq V returns \mathrm{Cut}(S), the total capacity of edges crossing (S,V\setminus S). The goal is to (i) compute a global minimum cut using subquadratic many queries; and as a key ingredient, (ii) compute an s-t maximum flow of value \tilde O(n) using only \tilde O(n^{5/3}) queries. (Throughout, \tilde O(\cdot) hides polylogarithmic factors.)

A second oracle we will simulate is the Bipartite-Independent-Set (BIS) query on (directed) residual graphs: given disjoint sets A,B, it asks whether there exists any edge from A to B. Crucially, a BIS query on an undirected graph can be simulated with 3 cut queries using
c(A,B) = \tfrac12(\mathrm{Cut}(A)+\mathrm{Cut}(B)-\mathrm{Cut}(A\cup B)).
Moreover, BIS queries on the residual graph G_f of an explicit flow f can also be simulated with 3 cut queries on G by subtracting the known contribution of f. This lets us explore residual neighborhoods and BFS layers using only cut queries.

We assume unit capacities unless noted; with integral capacities in [1,W] the bounds below scale by W.

2. Deterministic s-t max-flow in \tilde O(n^{5/3}) cut queries

2.1 Classical structure: Dinitz with blocking flows

Work in iterations. Maintain flow f, residual graph G_f, and the layered graph G_L given by BFS levels L_0=\{s\},L_1,\dots in G_f. Each iteration finds a blocking flow f’ in G_L that saturates at least one edge on every s-t path, augments f\leftarrow f+f’, and recomputes G_f. The key invariant is that after each iteration the layered distance strictly increases:
\mathrm{dist}_{G_f}(s,t)\ \text{increases by at least 1}.
With integral capacities \le W, Dinitz terminates in O(n^{2/3}W) iterations using the standard “short/long layered distance” analysis: once \mathrm{dist}\ge n^{2/3}, at most O(n^{2/3}W) further flow remains.

2.2 Implementing Dinitz with only cut queries

Two primitives suffice, both via BIS\tocut simulation:

 BFS / layered graph: Build a BFS tree of G_f from s in \tilde O(n) BIS queries (hence \tilde O(n) cut queries). The idea is to recover residual neighbors by repeated BIS-guided binary search on candidate sets; total cost charges to discovered vertices.

 Blocking flow search: Maintain a stack path s\leadsto t within the layered graph, pushing a next-layer neighbor if a BIS test finds one; otherwise pop and delete that vertex from the search. Every push/pop costs \tilde O(1). Each time the stack reaches length d’=\mathrm{dist}_{G_f}(s,t), one unit of flow is sent and the stack drops by d’. Over the whole blocking-flow phase this costs
 \tilde O\big(n + d’\cdot \mathrm{val}(f’)\big)
 BIS (thus cut) queries, where \mathrm{val}(f’) is the value of the blocking flow found in that iteration.

2.3 Two-phase query bound

Sum the per-iteration costs over all iterations:

 Phase I (short distance): While d’<n^{2/3}, the total additional flow sent is at most the final max-flow value, assumed \tilde O(nW). With at most n^{2/3} iterations and d’<n^{2/3},
 \sum \tilde O\big(n + d’\cdot \mathrm{val}(f’)\big)\ \le\ \tilde O(n\cdot n^{2/3})\ +\ \tilde O(n^{2/3}\cdot nW)\ =\ \tilde O(n^{5/3}W).

 Phase II (long distance): Once d’\ge n^{2/3}, only O(n^{2/3}W) more flow remains, and d’\le n. With at most n^{2/3} iterations,
 \sum \tilde O\big(n + d’\cdot \mathrm{val}(f’)\big)\ \le\ \tilde O(n\cdot n^{2/3})\ +\ \tilde O(n\cdot n^{2/3}W)\ =\ \tilde O(n^{5/3}W).

Therefore an s-t max-flow of value \tilde O(nW) can be computed deterministically in \tilde O(n^{5/3}W) cut queries; in unweighted simple graphs W=1, this is \tilde O(n^{5/3}).

3. From s-t max-flow to global min-cut in \tilde O(n^{5/3}) queries

The global min-cut routine follows a thresholded framework: given parameter \tau\le \delta-1, where \delta is the minimum degree, either (a) output a cut of size \le \tau or (b) certify the global min-cut >\tau, using \tilde O(n^{5/3}) queries. Then binary search on \tau (and the trivial \lambda=\delta case) produces the exact global min-cut.

Two core subroutines are needed in the cut-query model:

 Minimum isolating cuts for a set of terminals R.

 A terminal-aware expander decomposition that preserves small cuts.

A structural device makes both max-flow instances small: dominating sets are separated by every cut of size \le \delta-1.

3.1 Dominating-set separation and how to find one quickly

Let R\subseteq V be a dominating set. Then every cut of size \le \delta-1 hits both sides of R (“(\delta-1)-separated”). Intuition: if some side contained all of R, vertices on the other side would all need at least \delta cross edges by domination, contradicting a \le \delta-1 cut. This uses simplicity (no parallel edges).

Moreover, one can deterministically find a dominating set of size |R|=\tilde O(n/\delta) with only \tilde O(n) cut queries:

 Reduction rule: While some v has degree >\delta/2 in the current graph G’, add v to R and delete v\cup N_{G’}(v). Each application removes \Omega(\delta) vertices, so this happens at most O(n/\delta) times, and its query cost charges \tilde O(1) to each deleted neighbor by BIS-guided neighbor recovery.

 Low-degree phase: When all current degrees \le \delta/2, consider the bipartite graph between W_1=N_G(R) and the remaining W_2=V(G’). Average degree arguments plus binary-search via cut queries find a vertex v\in W_1 adjacent to \Omega(|W_2|\delta/n) many vertices of W_2; add v to R and delete its neighbors from G’. This shrinks G’ by a constant factor every O(n/\delta) steps, giving |R|=\tilde O(n/\delta). Total queries remain \tilde O(n).

This R ensures that all flow instances used later (isolating cuts, cut-matching) have total flow \tilde O(n) from the super-source/sink, so the \tilde O(n^{5/3}) max-flow routine applies.

4. Minimum isolating cuts via \tilde O(n^{5/3}) queries

Given terminals R and a threshold \tau, an isolating cut for r\in R is a minimum \{r\}–(R\setminus\{r\}) cut; the minimum isolating cut of R is the minimum over r\in R. The task is to either find one of size \le \tau or certify all isolating cuts >\tau, with |R|=\tilde O(n/\tau). The algorithm adapts a bit-decomposition trick so that only \tilde O(\log |R|) max-flow instances are solved, each of total value \tilde O(n).

Procedure.

 Encode & split: Assign each r\in R a distinct O(\log|R|)-bit label. For each bit position c, define a bipartition (A_c,B_c) of R by bit c. Every pair of terminals is separated in at least one bipartition.

 Auxiliary graph H: For a bipartition (A,B), create H by adding super-source s_{\text{src}} and super-sink s_{\text{snk}}; connect s_{\text{src}}\to a for a\in A and b\to s_{\text{snk}} for b\in B with capacity \tau+1. Edges of G have capacity 1. Compute an s_{\text{src}}{\text{–}}s_{\text{snk}} max-flow. (To keep the graph simple and unit-capacity, replace capacity \tau+1 by \tau+1 parallel subdivided unit-edges; this increases n only by \tilde O(n).)

 Record restricted min-cut: Let (C_A,C_B) be the min s_{\text{src}}{\text{–}}s_{\text{snk}} cut restricted back to G (remove the supers). By max-flow/min-cut, (C_A,C_B) is a min cut separating C_A\cap A from C_B\cap B in G.

 Delete and grow components: Delete \partial C_A (for all bit bipartitions). For each r\in R, let T_r be the vertices now reachable from r; put into R’ those r that were never saturated in any bipartition max-flow (formally: the super-source edge into r was not saturated whenever r\in A, and symmetrically for B). Then the sets T_r for r\in R’ are pairwise disjoint.

 Local contraction tests: For each r\in R’, contract V\setminus(\{r\}\cup(T_r\setminus R)) to a single sink s_r in a new graph G_r (keeping parallel edges). Compute the min (r,s_r) cut; if its size \le\tau, output it as an isolating cut; otherwise set \lambda_r=\infty. Because T_r are disjoint, the sum of the query costs over all r\in R’ is bounded by \sum \tilde O(|T_r|^{5/3})\le \tilde O(n^{5/3}).

Correctness. If some r^\star has a closest isolating cut C_{r^\star} of size \le\tau, then in every bipartition (A,B) that separates r^\star from some other terminal, r^\star is unsaturated, and C_{r^\star}\subseteq C_A (when r^\star\in A) by submodularity/closest-cut arguments. Hence C_{r^\star}\subseteq T_{r^\star}, so the contracted test in G_{r^\star} recovers a cut \le\tau. If no r has isolating cut \le\tau, the procedure certifies this.

Query complexity. Each bipartition’s max-flow in H has total flow value \tilde O(n) (all terminals live in R of size \tilde O(n/\tau), each incident capacity \le \tau+1); thus each is solvable in \tilde O(n^{5/3}) cut queries. There are only O(\log|R|)=\tilde O(1) bipartitions, and the disjoint-sum bound over local tests yields a total of \tilde O(n^{5/3}) queries.

5. Handling unbalanced vs. balanced small cuts

Let R be \tau-separated (e.g., a dominating set) and \varphi\in \mathrm{poly}(1/\log n).

 A cut C=(C_1,C_2) with |\partial C|\le \tau is \varphi-unbalanced for R if \min\{|C_1\cap R|,|C_2\cap R|\}\le (1/\varphi)^3+1/\varphi; otherwise it is \varphi-balanced.

5.1 Unbalanced case → isolating cuts suffice

Using a splitter family \mathcal{F} of size \tilde O(1) (deterministically constructed subsets with the property that every small subset of R is “hit” exactly once), one can run the isolating-cut routine on the sets in \mathcal{F}. If an unbalanced cut of size \le\tau exists, one of these calls will succeed and output a cut \le\tau. Total queries remain \tilde O(n^{5/3}).

5.2 Balanced case → terminal-aware expander decomposition

When all \le\tau cuts are \varphi-balanced for R, compute a decomposition of V into parts V_1,\dots,V_k such that each induced subgraph G[V_i] is a (\varphi,R_i)-almost expander with a large core R’i\subseteq R_i=R\cap V_i, and the number of inter-part edges is bounded by
\sum{i\ne j} |E(V_i,V_j)|\ =\ O\big(\varphi\,|R|\,(\tau+1)\,\log^6 n\big).
This is achieved via a cut-matching game with a strengthened matching player:

 In each round, the cut player bisects R (explicit, no queries).

 The matching player either returns a \varphi(\tau+1)-sparse balanced cut (then we are done), or embeds a (\tau+1)-regular almost perfect matching between the two sides using multicommodity flow with congestion 1/\varphi. To implement this with cut queries, add s_{\text{src}} to one side with capacity \tau+1 per terminal and s_{\text{snk}} to the other; edges of G get capacity 1/\varphi. The total flow is \tilde O(n), so each round costs \tilde O(n^{5/3}) queries.

After O(\log n) rounds, the union of perfect b-matchings has sparsity \Omega(\tau+1). Some edges are fake (to complete almost matchings); expander pruning removes a tiny set P of vertices so that the remaining graph (without fake edges) has conductance \tilde\Omega(1). Defining R’ = (R\setminus P) minus the few terminals with too many edges touching P or fake edges gives |R’|\ge (1-1/\log n)|R|. One concludes that each piece G[V_i] is a (\varphi,R_i)-almost expander with large core R’_i. All of this holds with \tilde O(n^{5/3}) total queries.

5.3 Sparsify the terminal set and recurse

Having the decomposition, build a smaller \tau-separated terminal set R_e by:

 Adding all R_i\setminus R’_i (few by construction),

 From each small part (very small core), pick one terminal from R’_i,

 From each large part, pick only 1+1/\varphi terminals from R’_i.

If no part has |\partial V_i|\le \tau, then |R_e|\ \le\ O\big(\varphi |R|\log^6 n\big)+ |R|/\log n, and one can show R_e remains \tau-separated. Choosing \varphi=1/\log^{10} n ensures |R_e|\le \tfrac12 |R|. Thus, either we already found a \le\tau cut (a part boundary) or we halve the terminal set while preserving separability. Depth is O(\log |R|)=\tilde O(1), and each level costs \tilde O(n^{5/3}) queries.

6. Threshold theorem and global min-cut

Threshold version. For any \tau\le \delta-1, there is a deterministic \tilde O(n^{5/3})-query algorithm that either outputs a cut of size \le\tau or certifies the global min-cut >\tau. This follows by:
(i) computing a \tilde O(n/\delta)-size dominating set R in \tilde O(n) queries;
(ii) if an unbalanced \le\tau cut exists, finding it via isolating cuts; otherwise (iii) performing the terminal-aware expander decomposition to obtain a smaller \tau-separated R_e and recursing; a \le\tau cut is found at some level or ruled out. Overall queries: \tilde O(n^{5/3}).

Global min-cut. Binary search over \tau (and handle the trivial \lambda=\delta case via a min-degree vertex) yields the exact global minimum cut in \tilde O(n^{5/3}) cut queries for simple unweighted graphs—the first deterministic subquadratic bound in this model.

7. Implementation notes and scope

 Unit capacities via subdivision. Super-source/sink edges of capacity \tau+1 are realized by \tau+1 parallel subdivided unit edges, keeping the graph simple and only \tilde O(n) larger. This preserves the \tilde O(n^{5/3}) query bounds of the max-flow routine.

 Why simplicity matters. The dominating-set separation lemma uses that every vertex in the “light” side of a small cut has degree \ge \delta with no parallel edges inflating capacity, so a \le \delta-1 cut cannot contain all of R on one side. Extending to weighted or non-simple graphs breaks this argument.

 Weighted graphs. With integer capacities in [1,W], the s-t routine costs \tilde O(n^{5/3}W) queries; the global-min-cut framework continues to apply where the specialized flow instances remain of value \tilde O(nW).

8. Complexity summary (deterministic, cut-query model)

 BIS simulation: 1 BIS on G_f \Rightarrow 3 cut queries on G. Neighborhood/BFS in \tilde O(n) queries.

 Blocking flow iteration: \tilde O\big(n + \mathrm{dist}\cdot \mathrm{val}(f’)\big) queries; total \tilde O(n^{5/3}W).

 Dominating set: size \tilde O(n/\delta) in \tilde O(n) queries.

 Isolating cuts: either find an isolating cut \le\tau or certify >\tau in \tilde O(n^{5/3}) queries (for |R|=\tilde O(n/\tau)).

 Expander decomposition (w.r.t. R): \tilde O(n^{5/3}) queries to obtain (\varphi,R)-almost expanders and \sum_{i\ne j}|E(V_i,V_j)|=O(\varphi|R|(\tau+1)\log^6 n).

 Global min-cut: \tilde O(n^{5/3}) queries overall.

1. The answer should clearly define the cut-query model, including how a query is made and what information it returns.
2. It should state the main goal: designing a deterministic algorithm for global min-cut using sub-quadratic queries, specifically ~O(n^(5/3)).
3. The trivial baseline of reconstructing the whole graph in ~O(n^2) queries should be mentioned as a contrast.
4. The solution must highlight that computing s–t max-flow of size ~O(n) with ~O(n^(5/3)) queries is the key subroutine.
5. The algorithm should be explained as an adaptation of Dinitz’s blocking flow algorithm to the cut-query setting.
6. The role of BIS queries and how they can be simulated with a constant number of cut queries should be described.
7. There should be an explanation of how to construct BFS layers in the residual graph using only ~O(n) cut queries.
8. The blocking flow procedure should be described, including its query complexity O(n + dist·val).
9. The two-phase complexity analysis (short distance < n^(2/3) and long distance ≥ n^(2/3)) should be included to justify the ~O(n^(5/3)) bound.
10. The answer should introduce the use of dominating sets, explain why every small cut must separate them, and how to compute such a set in ~O(n) queries.
11. It should define isolating cuts, explain the bit-partition encoding trick, and show how this reduces the problem to ~O(log|R|) max-flow computations.
12. The unbalanced vs balanced cases for small cuts should be distinguished, with isolating cuts handling the unbalanced case.
13. The balanced case should be resolved using an expander decomposition tailored to the terminal set, implemented via the cut-matching game and expander pruning.
14. The recursive framework that repeatedly halves the terminal set while preserving separability should be explained, ensuring depth O(log n).
15. Finally, the overall result should be summarized: the first deterministic sub-quadratic algorithm for global min-cut and efficient cut-query implementations for isolating cuts and expander decomposition.

The problem of computing the global min-cut of a graph, also known as edge connectivity, is a fundamental problem in algorithmic graph theory. This problem involves finding the smallest set of edges whose removal disconnects the graph. The study of this problem has been extensive, with various computational models being explored, including dynamic, parallel, distributed, and streaming algorithms. One model that has attracted significant attention recently is the cut-query model, where the edge set of the graph is unknown, and the algorithm can only make cut queries to an oracle that returns the number of edges crossing a given cut. Previous work in this model has focused on randomized algorithms, but the deterministic approach remains a significant challenge. The best-known deterministic algorithm for edge connectivity still requires a large number of queries, and there is a substantial gap between the upper and lower bounds for deterministic algorithms. | | **Global Min-Cut**: The smallest set of edges whose removal disconnects the graph. | | **Edge Connectivity**: The minimum number of edges that need to be removed to disconnect the graph. | | **Cut-Query Model**: A model where the algorithm can only make queries to an oracle that returns the number of edges crossing a given cut. | | **Deterministic Algorithm**: An algorithm that always produces the same output for a given input, without any randomness. | | **Sub-Quadratic Queries**: Queries that are less than quadratic in the number of vertices, i.e., \( \tilde{O}(n^{5/3}) \). | | **Expander Decomposition**: A technique to decompose a graph into expander graphs, which are graphs with good connectivity properties. | | **Isolating Cuts**: A technique to find cuts that isolate specific vertices or sets of vertices in the graph. | | **s-t Max-Flow**: The maximum flow that can be sent from a source vertex \( s \) to a sink vertex \( t \) in a flow network.

- **Cut Query**: A query to an oracle that returns the number of edges crossing a given cut in a graph. Specifically, for a set \( S \subseteq V \) in an undirected capacitated graph \( G \), the cut query oracle returns the total capacity of cut-edges \( \text{Cut}(S) = \sum_{a \in S, b \in \overline{S}} c(a, b) \). | | - **Bipartite Independent Set (BIS) Query**: A query to an oracle that returns whether there exists an edge between two disjoint sets \( A \) and \( B \) in a graph. For sets \( A, B \subseteq V \) in a capacitated graph \( H \), the BIS query oracle returns a boolean value indicating if \( c(A, B) = \sum_{a \in A, b \in B} c(a, b) > 0 \). | | - **Layered Graph**: Given an undirected graph \( G \) and a source vertex \( s \), the layered graph \( GL \) is a decomposition of the vertices of \( G \) into layers based on their distance from \( s \) in the residual graph \( Gf \). The \( i \)-th layer \( Li \) is defined as \( \{v \mid \text{dist}_{Gf}(s, v) = i\} \). | | - **Blocking Flow**: In the context of the layered graph \( GL \), a blocking flow \( f' \) is an \( s \)-\( t \) flow such that for every \( s \)-\( t \) path in \( GL \), some edge is saturated by \( f' \). | | - **(φ, R)-Expander**: A graph \( G \) is a (φ, R)-expander if for every cut \( (S, V \setminus S) \), the sparsity \( \Phi(S) = \frac{|∂S|}{\min\{|R \cap S|, |R \setminus S|\}} \geq φ(τ + 1) \). | | - **(φ, R)-Almost Expander**: A graph \( G \) is a (φ, R)-almost expander with core \( R' \) if \( R' \subseteq R \), \( |R'| \geq |R|(1 - \frac{1}{\log n}) \), and for any cut \( (S, V \setminus S) \), the sparsity \( \Phi(S) = \frac{|∂S|}{\min\{|R' \cap S|, |R' \setminus S|\}} \geq φ(τ + 1) \). | | - **Minimum Isolating Cut**: For any set of vertices \( R \subseteq V \) and \( r \in R \), the minimum isolating cut of \( r \) is an \( \{r\}-(R \setminus \{r\}) \) min-cut. The minimum isolating cut of \( R \) is the minimum-sized cut among the minimum isolating cuts over all \( r \in R \).

### Algorithm Improvement | - **Leverage Existing Algorithms**: Start by reviewing classic algorithms and think about how to modify them for new requirements. For example, use the Dinitz algorithm as a foundation. | - **Two-Phase Approach**: Divide the problem into two phases based on the shortest path distance. When the distance is less than or equal to \(n^{2/3}\), it's phase one; otherwise, it's phase two. Analyze and handle each phase separately. | | ### Problem Transformation and Simplification | - **Problem Conversion**: Turn the global min-cut problem into a Steiner min-cut problem. For a terminal set \(T\), a Steiner min-cut separates a pair of terminals in \(T\). If \(T = V(G)\), this becomes the global min-cut. | - **Use Dominating Sets**: Simplify the problem by choosing a dominating set \(D\) as the initial terminal set instead of the entire vertex set \(V(G)\). This set \(D\) has a size of \(\tilde{O}(n/\delta)\), where \(\delta\) is the minimum degree, and the global min-cut must separate \(D\). | | ### Isolating Cuts and Expander Decomposition | - **Isolating Cuts Technique**: Use isolating cuts to find the minimum isolating cut for specific vertices. | - **Expander Decomposition**: Implement expander decomposition and prove its effectiveness in the cut-query model. Use it as a tool for graph partitioning and decomposition, especially when balancing cut size and vertex count is important. | | ### Deterministic Algorithm Design | - **Deterministic Algorithms Benefits**: Look into converting randomized algorithms into deterministic ones to enhance reliability and applicability. | - **Graph Properties Utilization**: Make the most of the graph's simplicity and unweighted nature to boost algorithm efficiency. | | ### Other Query Models Reference | - **Compare Different Query Models**: Look into other query models for inspiration. For instance, study how matrix properties are learned via vector-matrix-vector queries or how graph connectivity is researched in quantum query models.


Improved Online Reachability Preservers

Question: How to construct a sparse reachability preserver online to preserve the reachability of given demand pairs in a directed graph?

Develop a framework in which to analyze online reachability preservers, and we use it to prove two new upper bounds, improving on both of the results mentioned above: | | **Theorem 1 (Online Pairwise Reachability Preservers).** Given an \( n \)-node directed graph \( G \), there is an online algorithm that constructs a reachability preserver \( H \) of \( |P| = p \) total demand pairs of size at most | \[ |E(H)| \leq O\left( \frac{n^{2+\alpha}}{3+\alpha + o(1)} p^{\frac{2}{3+\alpha}} + n^{2-2\alpha+o(1)} p^{\alpha} + n \right) < O\left( n^{0.72} p^{0.56} + n^{0.6} p^{0.7} + n \right), \] | where \( \alpha \geq 0.7 \) is a root of \( 4x^3 - 13x^2 + 10x - 2.1 \). | | **Theorem 2 (Online Source-Restricted Reachability Preservers).** Given an \( n \)-node directed graph \( G = (V, E) \), and a promise that the demand pairs will satisfy \( P \subseteq S \times V \) for some set of source nodes \( S \subseteq V \), there is an online algorithm that constructs a reachability preserver \( H \) of \( p \) total demand pairs of size at most | \[ |E(H)| \leq O\left( (n|S|p)^{1/2} + n \right). \] | The same result holds under the promise that \( P \subseteq V \times S \), but requires a slightly different algorithm. | | All of our construction algorithms are deterministic, run in polynomial time, and do not require any knowledge of \( P \) or \( S \) (including their size). **Theorem 2** ties the state-of-the-art upper bound in the offline sourcewise setting [1]. | | It may also be interesting to compare **Theorem 1** to the following lower bound, proved in [8] in a stronger online model: | | **Theorem 3.** Consider a stronger version of the online model where the graph \( G \) is initially empty, and an adversary may add new edges to \( G \) in each round before providing the next demand pair. Then there is a strategy for the adversary that guarantees that any online reachability preserver \( H \) will have size | \[ |E(H)| \geq \Omega\left( (np)^{2/3} + n \right). \] | | Our bound in **Theorem 1** is polynomially better than this one (in exchange for a weaker adversary), and so it formally separates these two models. | | A couple of the new tools that we develop for the online setting also turn out to be helpful in the classical offline setting. This yields the following small polynomial improvement in the state of the art: | | **Theorem 4 (Offline Reachability Preservers).** Given an \( n \)-node directed graph \( G \) and a set of demand pairs \( P \), one can construct in polynomial time a reachability preserver \( H \) of size | \[ |E(H)| \leq O\left( n^{\frac{3}{4}} p^{\frac{1}{2}} + n^{2 - \sqrt{2} + o(1)} p^{\frac{1}{\sqrt{2}}} + n \right) \leq O\left( n^{0.75} p^{0.5} + n^{0.59} p^{0.71} + n \right). \] | | **Theorem 5 (Non-Adaptive Reachability Preservers).** There are online algorithms satisfying **Theorem 1** and **Theorem 2** where the selected path for each demand pair \( (s, t) \) depends only on the input graph \( G \), the set of source nodes \( S \) (for **Theorem 2**), and | - the index \( i \) of the current demand pair, or | - the total number \( p \) of demand pairs that will arrive. | | **Theorem 6 (Offline UDSN [1, 15]).** There is a randomized polynomial time algorithm for (offline) UDSN that, given an \( n \)-node input graph \( G \) and a set of \( p \) demand pairs \( P \), returns a reachability preserver \( H \) of size | \[ |E(H)| \leq \text{OPT} \cdot O\left( \min\left\{ n^{\frac{4}{7} + \varepsilon}, p^{\frac{1}{2} + \varepsilon} \right\} \right) \] | where \( \text{OPT} \) is the number of edges in the sparsest possible reachability preserver of \( G, P \). | | **Theorem 7 (Online UDSN, parametrized on \( p \) [21]).** There is a randomized polynomial time algorithm for online UDSN that achieves a competitive ratio of \( O(p^{\frac{1}{2} + \varepsilon}) \). | | Thus, the remaining question is whether the dependence on \( n \) in **Theorem 6** can be recovered in the online setting. Here the state of the art is due to Grigorescu, Lin, and Quanrud, who obtained a competitive ratio of \( n^{\frac{2}{3} + \varepsilon} \) [24], roughly following the framework of Chlamtác et al. [16] but with adaptations for the online setting. As an application of our new online source-restricted preservers, we are able to substitute them into this framework, improving the competitive ratio: | | **Theorem 8 (Online UDSN, parametrized on \( n \)).** There is a randomized polynomial time algorithm for online UDSN that achieves a competitive ratio of \( O(n^{\frac{3}{5} + \varepsilon}) \). | | # Technical Preliminaries | ## The DAG reduction | While proving all of our upper bounds, it will be convenient to assume that the input graph is a DAG. That this assumption is without loss of generality comes from the following standard reduction, which we will recap somewhat briefly here: | | **Theorem 9 (DAG Reduction (folklore)).** If there is an algorithm that constructs a reachability preserver \( H \) of size | \[ |E(H)| \leq f(n, p, \sigma) \] | for any \( p \) demand pairs using \( |S| = \sigma \) source (or sink) nodes in an \( n \)-node DAG, then there is an algorithm that constructs a reachability preserver \( H \) of size | \[ |E(H)| \leq f(n, p, \sigma) + 2n \] | in arbitrary graphs. | | **Proof.** Given an input graph \( G \), compute a strongly connected component (SCC) decomposition. For each component \( C \) in the decomposition, choose an arbitrary vertex \( v \in C \) and an in- and out-tree rooted at \( v \) spanning \( C \). There are \( 2(|C| - 1) \) edges in these two trees, and hence there are slightly less than \( 2n \) edges in total, across all trees. | | We add all edges in these trees to our reachability preserver \( H \) at their first opportunity, and then for the rest of the construction we treat each SCC as a single contracted supernode, yielding a DAG \( G' \). For each demand pair \( (s, t) \), we can map the nodes \( s, t \) onto the corresponding supernodes in \( G' \) and choose paths in \( G' \) using our assumed DAG algorithm, to get a reachability preserver \( H' \) in \( G' \) on \( \leq f(n, p, \sigma) \) edges. Each edge \( (u, v) \) added to \( H' \) can be mapped back to any single edge in \( G \) between the set of nodes corresponding to the supernode \( u \) and the set of nodes corresponding to the supernode \( v \). Since our in- and out-trees preserve strong connectivity among all nodes in each of these sets, this will give a correct preserver in \( G \). | | ## Path System Definitions | A path system is a pair \( S = (V, \Pi) \), where \( V \) is a set of vertices and \( \Pi \) is a set of nonempty vertex sequences called paths. We will next recap some basic definitions; see also [8] for more discussion. | | Note that, even when the vertex set \( V \) is that of some graph \( G \), the paths in \( \Pi \) are abstract sequences of vertices that do not necessarily correspond to paths in \( G \). The length of a path \( \pi \in \Pi \), written \( |\pi| \), is its number of vertices (hence off by one from the length of the path through some graph). The degree of a node \( v \in V \), written \( \deg(v) \), is the number of paths that contain \( v \) (which may differ significantly from its degree as a node in a graph). The size of \( S \), written \( \|S\| \), is the quantity | \[ \|S\| := \sum_{\pi \in \Pi} |\pi|. \] | | We note that \( k \)-bridges still count even when they are degenerate; for example, two paths that coincide on two consecutive nodes count as a 2-bridge, and three paths that coincide on three consecutive nodes contain a 3-bridge, etc. | | A path system is **acyclic** if it does not contain any directed cycle as a subsystem, or equivalently, if there is a total ordering of the vertices \( V \) such that the order of vertices within each path \( \pi \in \Pi \) agrees with this ordering. We will frequently consider **ordered path systems**, which are path systems with a total ordering on their path set \( \Pi \). With this, we will sometimes only forbid subsystems with certain ordering constraints, e.g., 3-bridges where the last arc comes before the first in the ordering of \( \Pi \). | | # Online Reachability Preservers | ## Path Growth Algorithms | | A key tool in our online upper bounds will be the following two (very similar) path selection algorithms. We use these algorithms to generate a path \( \pi(s, t) \) for each demand pair \( (s, t) \) as it arrives, and then we add the edges of this path to the current preserver. Our path generation algorithms are greedy, growing paths one edge at a time and locally avoiding new edges if possible. | | ### Algorithm 1: Forwards-Growth Path Generation | ```python | Input: DAG \( G = (V, E) \), current preserver \( H \subseteq G \), demand pair \( (s, t) \) | 1. Let \( \pi \leftarrow (s) \) | 2. while last node of \( \pi \) is not \( t \) do | 3. \( u \leftarrow \) last node of \( \pi \) | 4. if there exists an edge \( (u, v) \in E(H) \) with \( t \) reachable from \( v \) then | 5. append any such edge \( (u, v) \) to the back of \( \pi \) | 6. else | 7. append to the back of \( \pi \) any edge \( (u, v) \in E(G) \) with \( t \) reachable from \( v \) | 8. return \( \pi \) | ``` | | ### Algorithm 2: Backwards-Growth Path Generation | ```python | Input: DAG \( G = (V, E) \), current preserver \( H \subseteq G \), demand pair \( (s, t) \) | 1. Let \( \pi \leftarrow (t) \) | 2. while first node of \( \pi \) is not \( s \) do | 3. \( v \leftarrow \) first node of \( \pi \) | 4. if there exists an edge \( (u, v) \in E(H) \) with \( u \) reachable from \( s \) then | 5. append any such edge \( (u, v) \) to the front of \( \pi \) | 6. else | 7. append to the front of \( \pi \) any edge of the form \( (u, v) \in E(G) \) with \( u \) reachable from \( s \) | 8. return \( \pi \) | ``` | | The two algorithms are symmetric to each other, and differ only in whether we grow the path from front to back or from back to front. | | As we use these algorithms to sequentially generate paths and build our preserver, it will be helpful to track an auxiliary path system \( Z = (V, \Pi) \). Each time we add a path \( \pi(s, t) \) to \( H \), say that a new edge is an edge \( e \in \pi(s, t) \) that was not previously in the preserver. We then add a corresponding path \( \pi' \) to \( Z \), whose nodes are | \[ \pi' := \begin{cases} | \{u \mid \text{there is a new edge } (u, v) \in \pi(s, t)\} \cup \{t\} & \text{if forwards-growth is used} \\ | \{v \mid \text{there is a new edge } (u, v) \in \pi(s, t)\} \cup \{s\} & \text{if backwards-growth is used} | \end{cases} \] | and in either case, these nodes are ordered in the path \( \pi' \) the same as their order in \( \pi(s, t) \). We will also treat \( Z \) as an ordered path system, with the paths in \( Z \) ordered by the arrival of the demand pairs that generated each path. The following properties of \( Z \) all follow straightforwardly from the construction: | | **Lemma 10 (Properties of \( Z \)).** | 1. \( Z \) is acyclic, | 2. \( \|Z\| = |E(H)| + p \), | 3. Under forwards-growth, \( Z \) has no bridge in which the first arc comes before the river in the ordering of \( \Pi \). Under backwards-growth, \( Z \) has no bridge in which the last arc comes before the river in the ordering of \( \Pi \). | | **Proof.** | 1. Since the input graph \( G = (V, E) \) is a DAG, the order of nodes in each path \( \pi \in \Pi \) agrees with the topological ordering of the nodes in \( V \), implying that \( Z \) is acyclic. | 2. Initially, we have \( \|Z\| = |E(H)| = 0 \). Then, every path \( \pi' \) added to \( Z \) corresponds to a path \( \pi(s, t) \) that contributes exactly \( |\pi'| - 1 \) new edges to \( H \), so in the end we have \( \|Z\| = |E(H)| + p \). | 3. We will prove this for forwards-growth; the argument for backwards-growth is symmetric (up to reversal of direction of the edges of the input graph \( G \)). Seeking contradiction, suppose there is a bridge formed by nodes \( (x_1, \ldots, x_k) \), arc paths \( \pi_1, \ldots, \pi_{k-1} \), and river path \( \pi_r \), with \( \pi_1 <_\Pi \pi_r \). Let \( \pi(s_1, t_1), \pi(s_r, t_r) \) be the paths generated by forwards-growth corresponding to \( \pi_1, \pi_r \) respectively. By construction, since \( x_1 \in (\pi_1 \cap \pi_r) \), these paths both contribute new edges to \( H \) leaving \( x_1 \); call the first one \( (x_1, y) \in \pi(s_1, t_1) \). Now notice that the arcs witness reachability among all of the node pairs | \[ | (y, x_2) \in \{z \mid z \in \pi_1\}, (x_2, x_3) \in \{z \mid z \in \pi_2\}, \ldots, (x_{k-1}, x_k) \in \{z \mid z \in \pi_{k-1}\}, (x_k, t_r) \in \{z \mid z \in \pi_k\}. | \] | So by transitivity, the node pair \( (y, t_r) \) is reachable. When we generate \( \pi(s_r, t_r) \) using forwards-growth, since we have already added \( \pi(s_1, t_1) \), the edge \( (x_1, y) \) is already present in \( H \) and we have \( (y, t_r) \) reachability. So the algorithm will not choose to add a new edge leaving \( x_1 \) while generating \( \pi(s_r, t_r) \), which completes the contradiction. | | | | | ## Online Source-Restricted Reachability Preservers | We will prove the following upper bound on source-restricted preservers in the online model: | | **Theorem 11.** In the online model with an \( n \)-node input DAG \( G = (V, E) \) and \( p \) total demand pairs, the final preserver \( H \) will have size | \[ |E(H)| \leq O\left( (np|S|)^{1/2} + n \right) \] | in either of the following two settings: | - \( S \) is the set of start nodes used by the given demand pairs \( P \) (that is, \( P \subseteq S \times V \)), and the builder generates paths in each round using the backwards-growth algorithm, or | - \( S \) is the set of end nodes used by the given demand pairs \( P \) (that is, \( P \subseteq V \times S \)), and the builder generates paths in each round using the forwards-growth algorithm. | | Notably, neither the forwards- nor backwards-growth algorithm requires the builder to know the number of demand pairs \( p \) or any information about the set of source/sink nodes \( S \). We will only prove the latter point in **Theorem 11**, analyzing forwards-growth and assuming \( P \subseteq V \times S \). The other point is symmetric. | | The proof will work by analyzing the path system \( Z \) associated with the online path-adding process; recall its essential properties in **Lemma 10**. Let \( \ell := \|Z\| / p \) be the average path length and let \( d := \|Z\| / n \) be the average node degree. If \( d \leq O(1) \) then we have \( \|Z\| \leq O(n) \) and the theorem holds, so we may assume in the following that \( d \) is at least a sufficiently large constant. | | Imagine that we add the paths from \( Z \) to an initially-empty system, one at a time, in the reverse of their ordering in \( \Pi \). We observe: | | **Lemma 12.** There is a path \( \pi \in \Pi \) such that when \( \pi \) is added in the above process, it contains at least \( \ell / 4 \) nodes of degree at least \( d / 4 \) each. | | **Proof.** Suppose not. Then, by counting the first \( d / 4 \) times each node appears in a path separately from the remaining times, the total size of \( Z \) can be bounded as | \[ \|Z\| \leq \frac{nd}{4} + \frac{p\ell}{4} \leq \frac{\|Z\|}{4} + \frac{\|Z\|}{4} = \frac{\|Z\|}{2}, \] | which is a contradiction. | | **Lemma 13.** \( \ell d \leq O(|S|) \). | | **Proof.** Suppose for contradiction that \( \ell d > 16|S| \). By the previous lemma, there is a path \( \pi \in \Pi \) that intersects at least \( \ell d / 16 > |S| \) other paths in \( \Pi \), which were added to the system before \( \pi \) (and hence come later than \( \pi \) in the ordering of \( \Pi \)). By the Pigeonhole principle, and since the demand pairs satisfy \( P \subseteq V \times S \), at least two of these intersecting paths \( q_1, q_2 \) end at the same node \( t \in S \). Since by **Lemma 10** \( Z \) does not contain any 2-bridges, \( q_1, q_2 \) may not intersect at any other nodes, and so they intersect \( \pi \) at two different nodes. But this implies that \( \pi, q_1, q_2 \) form a 3-bridge in which \( \pi \) is the first arc and it precedes both \( q_1, q_2 \) in the ordering of \( \Pi \) (see Figure 2). This contradicts **Lemma 10**, completing the proof. | | We now complete the proof by algebraically rearranging the inequality from the previous lemma. We have: | \[ \ell d \leq O(|S|) \] | \[ (p\ell)(nd) \leq O(|S|pn) \] | \[ \|Z\|^2 \leq O(|S|pn) \] | \[ \|Z\| \leq O((|S|pn)^{1/2}) \] | | Since by **Lemma 10** we have \( \|Z\| \geq |E(H)| \), this implies our desired bound on the size of the output preserver. | ## Online Pairwise Reachability Preservers | | By following an identical proof strategy to our upper bound in the source-restricted setting (i.e., exploiting forbidden ordered 2- and 3-bridges), it is possible to prove an upper bound of |E(H)| ≤ O((np)^(2/3) + n) (details omitted, since we will show a stronger bound than this). As discussed in [8], this is probably the best upper bound one can show by exploiting only the forbidden ordered 2- and 3-bridges from Lemma 10. Nonetheless, we will show a stronger bound, which crucially also exploits the forbidden ordered 4-bridges from Lemma 10. | | Recap of [8]. Recent work of Bodwin, Hoppenworth, and Trabelsi [8] on offline reachability preservers introduced a framework for extremal analysis of forbidden 4-bridges, which we will briefly recap here. First: | **Lemma 14 (Independence Lemma, c.f. [8], Lemma 38).** Let β(n, p, ∞) denote the maximum possible size of a path system with n nodes, p paths, and no bridges as subsystems. Then every n-node directed graph and set of p demand pairs has an offline reachability preserver H of size |E(H)| ≤ O(β(n, p, ∞)), and this is asymptotically tight. | | Thus, it suffices to argue about the extremal size of a bridge-free path system. We remark here that versions of this lemma are perhaps implicit at a low level in prior work. It is also an inherently offline lemma, and breaks down completely in the online setting; the reliance on this lemma (or its underlying ideas) is essentially why the known results for offline reachability preservers do not tend to extend to the online setting. | | This previous paper then argues as follows. Assume for convenience that all paths have length Θ(ℓ) and all nodes have degree Θ(d), for some parameters ℓ, d. Recall that the upper bound from forbidden 2- and 3-bridges is O((np)^(2/3) + n), and so our goal is to show that this bound cannot be tight. A few straightforward calculations reveal that, if this bound were tight, then for the typical pair of paths π1, π2 in the system there will be Θ(ℓ) paths that intersect π1 and then π2. However, no pair of these intersecting paths may have crossing intersection points with π1, π2. | | If there are Θ(ℓ) paths that intersect both π1, π2, and yet these intersecting paths cannot cross each other, then the typical intersecting path must “lie flat” in the sense that there is not much of a gap along either π1 or π2 to, say, the h nearest intersecting paths (for some parameter h). In order to exploit this, the key strategy in [8] is to sample a random base path πb ∈ Π, and then analyze the random subsystem S′ on the vertex set formed by examining the h adjacent nodes along the paths that intersect πb. | | Since πb extends along its branching paths, we should expect S′ to contain many long paths. But we can apply known upper bounds to S′ to rule out this possibility. This implies that, in fact, the typical pair of paths π1, π2 have ≪ ℓ paths that intersect both, leading to an improved upper bound. | | Our Offline Improvements. An auxiliary result of this paper is an improvement in the bound shown by [8]. We refer back to Theorem 4 for the statement, or Appendix B for the proof. The source of these improvements is from an improved strategy for controlling the size of the random subsystem S′. One of the two ways in which this part improved is by recursively bounding the size of S′ (this is executed in Lemma 40). Although this idea is conceptually straightforward, it requires a significant refactoring of the proof to enable it. The technical reason is that [8] bounds the input system Z using an ℓ1 norm of path lengths (the standard notion of size) but S′ using an ℓ2 norm of path lengths, making it impossible to directly apply the bound on Z recursively to S′. We switch to bounding both using the ℓ2 norm everywhere, and we only move back to our desired ℓ1 norm at the very end of the proof. The other new ingredient is an improved counting of the contribution of “short” paths to the size of S′, which is executed in Lemma 40. | | Our Online Adaptation. It will be slightly more convenient in this exposition to consider the backwards-growth strategy for path generation here, although of course by symmetry either strategy works (we use this convention in Appendix A as well). We will analyze the path system Z constructed above, and in particular Lemma 10 states that bridges are forbidden in Z whose last arc comes before the river. | | Can we exploit forbidden ordered 4-bridges by following the strategy outlined above? Some parts of the method extend easily, with minor tweaks. For example, instead of counting any paths π that intersect the typical pair of paths π1, π2, we can restrict attention to those that also come after π1, π2 in the ordering. Then the “crossing” in Figure 3 is still forbidden (since the river π3 is assumed to come after the last arc π2). Relatedly, when we define S′, we need to consider only the paths that intersect πb and come before it. Nonetheless, all these definitional adaptations turn out to affect the relevant counting arguments by only constant factors, and so they do not harm the argument. There are many other minor issues that we will not overview, which can be dispatched with a little technical effort. | | However, there is one major problem: there is now potential for overlap in the paths that branch off πb. That is, in the offline setting, we can exploit 3-bridge-freeness to argue that no two paths that branch off πb also intersect each other, and thus every node in S′ (except those on πb itself) is in exactly one branching path. But in the online setting, this is not so: these intersections form a 3-bridge that may be allowed. | | Naively, the typical node in S′ could be in Θ(d) branching paths, and this complication completely wipes out all the gains from the analysis. Although we cannot rule out the possibility of some nodes having Θ(d) branching paths, we are able to use a more intricate maneuver to show that there is some threshold t ≪ d such that, in expectation, the number of nodes of branching degree Ω(x) in S′ is far less than the trivial bound of ℓdh/x for x ≥ t. This turns out to be good enough to recover some gains from the method. We unfortunately cannot show this for t = 1, which is the fundamental reason why our online bounds are polynomially worse than the corresponding offline bounds. | | # Non-Adaptive Reachability Preservers | We next describe our method to convert our online algorithms to (almost) non-adaptive algorithms. Our algorithm essentially works by simulating a greedy adversary in the online model, who repeatedly provides the most costly demand pair in each round, and then we set paths by running our online algorithm against this adversary, halting at the appropriate place. Our proof will thus imply indirectly that this greedy strategy is the most effective one for an adversary in the online model, up to constant factors. | | Our algorithms will reference an extremal function *f*(*n*, *p*) for online reachability preservers, achieved by a generic path selection algorithm π(*s*, *t* | *G*, *H*). That is, we imagine an algorithm that starts with *G* as any *n*-node graph, and *H* as the *n*-node empty graph. When each demand pair (*s*, *t*) arrives, we select the path π(*s*, *t* | *G*, *H*) and add all of its edges to *H*. Then *f*(*n*, *p*) is the largest possible number of edges in *H* after *p* rounds of this process. | | We use this generic extremal function *f*, rather than the particular extremal upper bound from Theorem 1, in order to emphasize that this bound and the technical details of the forwards- or backwards-growth algorithms are not really important in this proof. If a future result improves on Theorem 1, then the new bound will automatically transfer to these results as well. For simplicity we will focus on *f*(*n*, *p*) here, but the proof would generalize readily to extremal functions that incorporate additional parameters beyond *n* and *p*. This includes the online source-restricted preservers of Theorem 11, which incorporate |*S*| as a parameter, although we note that these require the set *S* to be given on input, so that we know the set of possible demand pairs and we can properly simulate the adversary (i.e., search over the proper subset of demand pairs in the condition of the while loop). | | **Input:** *n*-node directed graph *G* = (*V*, *E*), number of demand pairs *p* | - All reachable node pairs (*s*, *t*) in *G* have “unfinalized” path | - *H* ← (*V*, ∅) | - Let *f*(*n*, *p*) be an extremal function for online reachability preservers, achieved by a deterministic path selection algorithm π(*s*, *t* | *G*, *H*) | - while there is reachable (*s*, *t*) with > *f*(*n*, *p*)/*p* edges in π(*s*, *t* | *G*, *H*) \ *E*(*H*) do | - finalize path π(*s*, *t* | *G*, *H*) for (*s*, *t*) | - add edges of π(*s*, *t* | *G*, *H*) to *H* | - foreach remaining unfinalized reachable pair (*s*, *t*) do | - finalize path π(*s*, *t* | *G*, *H*) for (*s*, *t*) | | **Algorithm 3: known-p-non-adaptive-rps** | | **Theorem 15.** For all *n*-node graphs *G* and sequences *P* of |*P*| =: *p* demand pairs, the paths set by Algorithm 3 satisfy | \[ | \left| \bigcup_{(s,t) \in P} \pi(s, t) \right| \leq 2f(n, p). | \] | | *Proof.* Let *Q* be the set of demand pairs whose paths are set in the initial while loop, and let *H*_*Q* be the subgraph *H* just after the paths for *Q* have been set and the while loop terminates. We first note that |*Q*| < *p*, since otherwise by counting the edges contributed to *H*, the first *p* demand pairs in *Q* create a subgraph *H*_*Q* of size |*E*(*H*_*Q*)| > *f*(*n*, *p*) which contradicts the definition of the extremal function *f*. Since |*Q*| < *p*, we therefore have | \[ | |E(H_Q)| \leq f(n, p). | \] | Meanwhile, all demand pairs in *P* \ *Q* have their path set in the final for loop, and by construction there are ≤ *f*(*n*, *p*)/*p* edges outside *H*_*Q* in each path. So we have | \[ | \left| \bigcup_{(s,t) \in P} \pi(s, t) \right| \leq |E(H_Q)| + \sum_{(s,t) \in P \setminus Q} |\pi(s, t) \setminus E(H_Q)| \leq f(n, p) + p \cdot \left( \frac{f(n, p)}{p} \right) = 2f(n, p). | \] | | This theorem implies that, if one uses the precomputed paths from Algorithm 3 to respond to online queries, then the online upper bound of *f*(*n*, *p*) will still apply (up to a factor of 2). The main weakness of Algorithm 3 is that it requires advance knowledge of the parameter *p*, in order to compute the threshold *f*(*n*, *p*)/*p* at which we exit the initial while loop. It is tempting, but incorrect, to think this can be generally avoided by setting all paths as in the main while loop. Such a strategy would work for a path selection algorithm that happens to satisfy an axiom like monotonicity, for which adding edges to *H* can only decrease the number of new edges in a selected path π(*s*, *t* | *G*, *H*) (it might also be fine to tolerate an approximate version of monotonicity). However, we note that the path selection algorithms (forwards- and backwards-growth) used in our online upper bounds are not monotonic in this way (or even approximately monotonic). | | That said, we next describe a wrapper for the algorithm that can avoid the need to know *p* ahead of time: | | **Theorem 16.** Suppose that the extremal function *f*(*n*, *p*) depends polynomially on its second parameter *p* in the regime where *p* ≥ *p*⁺. Then the online algorithm that runs Algorithm 4 as a preprocessing routine upon receiving *G*, and which then uses Algorithm 5 to select the path added to the preserver for each *i*ᵗʰ demand pair (*s*, *t*), will construct a reachability preserver *H* of size |*E*(*H*)| ≤ *O*(*f*(*n*, *p*)). | | **Input:** *n*-node directed graph *G* = (*V*, *E*) | - Let *f*(*n*, *p*) be an extremal function for online reachability preservers, achieved by a deterministic path selection algorithm π(*s*, *t* | *G*, *H*) | - Let *p*⁺ := arg maxₚ *f*(*n*, *p*) ≤ *O*(*n*) | - foreach *q* ∈ {*p*⁺, 2*p*⁺, 4*p*⁺, 8*p*⁺, . . . } do | - Run Algorithm 3 with number-of-paths parameter *q* | - Denote selected paths by π_*q*(*s*, *t*) | | **Algorithm 4: Preprocessing for Index-Sensitive Non-Adaptive Reachability Preservers** | | **Input:** demand pair (*s*, *t*), index *i* | - // run Algorithm 4 as preprocessing | - Let *q* be the least value in {*p*⁺, 2*p*⁺, 4*p*⁺, 8*p*⁺, . . . } with *q* ≥ *i* | - Return π_*q*(*s*, *t*) | | **Algorithm 5: Path Selection for Index-Sensitive Non-Adaptive Reachability Preservers** | | *Proof.* For each possible choice of *q*, we will add at most *q* paths selected by π_*q* to the preserver. By Theorem 15, these paths will have at most 2*f*(*n*, *q*) edges in their union. Additionally, letting *q*⁺ be the largest choice of *q* for which we add any corresponding paths, note that we have *q*⁺ ≥ *p* ≥ *q*⁺/2. So we can bound the total number of edges in the preserver as: | \[ | |E(H)| \leq \sum_{q \in \{p^+, 2p^+, 4p^+, 8p^+, \ldots, q^*\}} 2f(n, q) \leq O(f(n, q^*)) \leq O(f(n, p)). | \] | Here the second inequality holds because *f* depends polynomially on its second parameter, and so this sum is asymptotically dominated by its largest term. The third inequality holds because we have *p* ≥ *q*⁺/2, and (again since *f* depends polynomially on its second parameter) this means the values of *f*(*n*, *q*⁺), *f*(*n*, *p*) differ by at most a constant factor. | | # Online Unweighted Directed Steiner Forest Algorithms | We will next apply our extremal bounds for online reachability preservers to the problem of Online UDSF. As a reminder, in this problem we receive an n-node directed graph G = (V, E) on input, and then in each round we receive a new demand pair (s, t) that is reachable in G. We must irrevocably add edges to a reachability preserver H to ensure that (s, t) is reachable in H before the next demand pair is received. We do not know the number of demand pairs p ahead of time. We will denote by OPT the size of the smallest possible (offline) reachability preserver for G, P, where P is the set of all demand pairs received. | ## Recap of the Grigorescu-Lin-Quanrud Bound | Our new bound will use the structure and several technical ingredients from the previous state-of-the-art online algorithm by Grigorescu, Lin, and Quanrud [24]. They proved: | | **Theorem 17 ([24]).** For online UDSF, there is a randomized polynomial time algorithm with competitive ratio *O*(*n*^(2/3+ε)). | | Their algorithm carries two parameters, *T* and τ, which will be set at the end by a balance. In the following, we will say that a demand pair (*s*, *t*) is nontrivial if it is not already reachable when it arrives, and thus it requires us to add at least one new edge to the preserver. We let *p* be the total number of nontrivial demand pairs. | | - For the first *T* nontrivial demand pairs that arrive, we use the following result by Chakrabarty et al.: | **Theorem 18 ([21]).** There is a randomized polynomial-time online algorithm that constructs a preserver of the first *T* demand pairs of size at most *OPT* · *O*(*T*^(1/2−ε)). |

1. The answer should explain what a reachability preserver is, and clarify the online model where demand pairs arrive one by one.
2. It should present Theorem 1 with the precise upper bound formula, including the dependence on n, p, and α, and note that α ≈ 0.7 is the root of a cubic.
3. It should present Theorem 2 (source-restricted preservers), giving the O((n|S|p)^(1/2) + n) bound and noting symmetry between source-restricted and sink-restricted cases.
4. It should emphasize that the algorithms are deterministic, polynomial-time, and require no knowledge of P or S in advance.
5. Theorem 3 should be included, describing the stronger adversarial model with edge insertions and the lower bound Ω((np)^(2/3)+n).
6. The answer should explicitly compare Theorem 1 to Theorem 3 and note that the new upper bound is polynomially better, thus separating the models.
7. It should mention that the new tools developed for online analysis also yield improvements in the offline setting, leading to Theorem 4 with the refined bound.
8. Theorem 5 should be included, describing the existence of non-adaptive online algorithms where paths depend only on G, S, and either the index i or total p.
9. Theorem 6 should be presented, stating the offline UDSN bound involving OPT and the min{n^(4/7+ε), p^(1/2+ε)} factor.
10. Theorem 7 should be mentioned, giving the online UDSN competitive ratio O(p^(1/2+ε)).
11. Theorem 8 should be included, improving the online UDSN bound parametrized on n to O(n^(3/5+ε)), and contrasted with the previous O(n^(2/3+ε)).
12. The DAG reduction (Theorem 9) should be summarized, explaining why analysis can assume input graphs are DAGs at the cost of adding ≤2n edges.
13. The answer should describe the path-growth algorithms (forwards and backwards), how they greedily select paths, and how they are used to build the preserver.
14. It should define the auxiliary path system Z, state its properties (acyclic, |Z| = |E(H)|+p, forbidden bridges), and explain how these are central to the analysis.
15. Finally, the overall contribution should be summarized: a unified framework for analyzing online reachability preservers, new upper bounds (Theorems 1–2), separation from lower bounds (Theorem 3), improvements in offline bounds, and new results for UDSN (Theorems 6–8).

## Definitions of Related Concepts | | 1. **Reachability Preservers**: | - Given a directed graph \( G = (V, E) \) and a set of demand pairs \( P \subseteq V \times V \), a reachability preserver is a subgraph \( H \subseteq G \) such that for all \( (s, t) \in P \), there is an \( s \leadsto t \) path in \( H \) if and only if there is one in \( G \). | | 2. **Offline Model**: | - In the offline model, both the graph \( G \) and all the demand pairs \( P \) are provided as input, and the goal is to construct a sparse reachability preserver \( H \). | | 3. **Online Model**: | - In the online model, the graph \( G \) is given, and the demand pairs \( (s, t) \in P \) arrive one by one. For each arriving demand pair, edges must be irrevocably added to the preserver \( H \) to ensure reachability for that pair before the next one arrives, without knowledge of future demand pairs. | | 4. **Competitive Ratio**: | - In online algorithms, the competitive ratio is the ratio of the size of the preserver constructed by the online algorithm to the size of the optimal offline solution (constructed with full knowledge of all demand pairs). The goal is to minimize this ratio. | | ## Research Background | | 1. **Applications of Reachability Preservers**: | - Reachability preservers are a fundamental graph sparsification tool with applications in various areas such as graph spanners, shortcut sets, property testing algorithms, flow/cut approximation algorithms, and Steiner network design algorithms. | | 2. **Historical Context of Reachability Preservers**: | - The study of reachability preservers dates back to the Directed Steiner Network (DSN) problem, a classic NP-hard problem. DSN aims to find the sparsest (or minimum-weight) reachability preserver for a given instance \( G, P \). | - Recently, there has been intensive research on reachability preservers from an extremal perspective, focusing on determining the worst-case number of edges needed as a function of the number of nodes \( n \) and the number of demand pairs \( p \). | | 3. **Challenges in the Online Setting**: | - The online setting for reachability preservers poses significant challenges as the algorithm must make irreversible decisions without knowledge of future demand pairs. This contrasts with the offline setting where the algorithm can leverage complete information about all demand pairs to construct an optimal solution. | | 4. **Previous Results**: | - **Coppersmith and Elkin**: Their work implies an online upper bound of \( O(\min\{np^{0.5}, n^{0.5}p\} + n) \). | - **Source-Restricted Setting**: When demand pairs are restricted to \( P \subseteq S \times V \) (or \( P \subseteq V \times S \)), the online upper bound for the number of edges is \( O(n|S|) \).

## Half-Bridge-Free System | A system is half-k-bridge-free if it contains no bridges of size at most k where the last arc comes before the river.

1. **Online Path Generation**: Dynamically maintain paths via local greedy strategies (Forward/Backward-Growth) to avoid dependency on global information. | 2. **Extremal Graph Theory Analysis**: Exploit combinatorial properties (bridge-freeness) of path systems and recursive techniques to bound edge counts. | 3. **Parameterization and Balancing**: Achieve tight upper bounds through optimized parameters (e.g., $ \ell, d, h $) and classified handling (source constraints, thick/thin pairs). | 4. **Application Transformation**: Integrate preserved graph results into online algorithm frameworks to enhance competitive ratios.


Prophet Secretary and Matching: the Significance of the Largest Item

Question: # Design and Analysis of Improved Online Algorithms for Prophet Secretary and Prophet Secretary Matching Problems | | ## Objectives | | ### Prophet Secretary Problem | Design an algorithm with a higher competitive ratio to break the previous 0.675 barrier. | | ### Prophet Secretary Matching Problem | Develop an algorithm to surpass the 1−1/e (approximately 0.632) competitive ratio barrier.

Use the continuous‐time formulation of prophet secretary. There are n items with non-negative values v_1,\dots,v_n drawn independently from F_1,\dots,F_n. WLOG we (i) discretize each F_i (continuous can be \varepsilon-approximated), and (ii) assume disjoint supports via infinitesimal tie-breaking so the maximum is unique. Let p_i^v=\Pr[v_i=v]. | | Each item i arrives at time t_i\sim\mathrm{Unif}[0,1] independently. Upon arrival we observe (i,v_i,t_i) and must accept or reject irrevocably; at most one item can be accepted. Performance is measured against the prophet \mathbb{E}[\max_i v_i]. An algorithm is \Gamma-competitive if its expected value is \ge \Gamma\cdot \mathbb{E}[\max_i v_i]. | | Equivalence to random order. The continuous model is equivalent to sampling a uniform random permutation and revealing items in that order; we work in continuous time because it yields cleaner integrals. | | Activation-Based Online Algorithms | | We represent any algorithm by acceptance/activation probabilities g_i^v(t)\in[0,1]: when item i with value v arrives at time t, we activate it with probability g_i^v(t) and accept the first activated item. | | It is convenient to parameterize g via activation rates a_i^v(t)\ge0: | g_i^v(t)\;=\;a_i^v(t)\cdot \exp\!\Big(-\!\int_0^t A_i(x)\,dx\Big), | \qquad | A_i(t)\stackrel{\rm def}{=}\mathbb{E}_{v\sim F_i}[a_i^v(t)]. | Under this form (a nonhomogeneous Poisson view): | • Probability item i activates before \theta: 1-\exp\!\big(-\!\int_0^\theta A_i(t)\,dt\big). | • Probability we accept item i with value v: | \int_0^1 p_i^v\,a_i^v(t)\cdot \exp\!\Big(-\!\int_0^t \sum_j A_j(x)\,dx\Big)\,dt. | | Let | \rho_i^v\stackrel{\rm def}{=} \prod_{j\ne i}\Pr[v_j< v], \qquad | x_i^v\stackrel{\rm def}{=} p_i^v\,\rho_i^v, | so \sum_{i,v} x_i^v=1. A sufficient condition for \Gamma-competitiveness is | \int_0^1 p_i^v\,g_i^v(t)\prod_{j\ne i}\Big(1-\!\int_0^t \mathbb{E}_{u\sim F_j}[g_j^u(x)]\,dx\Big)\,dt\;\ge\; \Gamma\,x_i^v | \quad(\forall i,v). | \tag{★} | | Warm-ups | • Constant Activation Rates a_i^v(t)=\rho_i^v for all t is (1-1/e)-competitive (classical). | • Two-stage Step Rates: split [0,1] at \beta and let pre/post rates sum to 2\rho_i^v; under a standard small-items condition this yields 0.694. | | Main Idea: Treat the Largest Item Separately | | Let the largest item i_0=\arg\max_i \sum_v x_i^v with x_0=\sum_v x_{i_0}^v. We focus on the item the prophet is most likely to take and give it a special (probability-based) policy, while all other items use rate-based step policies with invariants that depend on i_0’s characteristics. This “one large item + many small” split is key to beating blind-strategy barriers. | | Algorithm: Step Activation Except One Large Item | | Parameters \beta_0\le \beta_1\le \beta_2 (chosen by computer-assisted optimization). | • Largest item i_0 (use probabilities, not rates): | g_{i_0}^v(t)= | \begin{cases} | z_{i_0}^v,& t\in[0,\beta_0),\\ | 0,& t\in[\beta_0,\beta_2),\\ | 2\rho_{i_0}^v - z_{i_0}^v,& t\in[\beta_2,1], | \end{cases} | \quad z_{i_0}^v=(2\rho_{i_0}^v-1)_+. | • Other items i\ne i_0 (use rates): | a_i^v(t)= | \begin{cases} | z_i^v,& t\in[0,\beta_1),\\ | 2\rho_i^v - z_i^v,& t\in[\beta_1,1], | \end{cases} | \quad \text{with } z_i^v\in\big[(2\rho_i^v-1)+,\;\rho_i^v\big] | and the invariant | \sum{i\ne i_0}\sum_v p_i^v z_i^v | \;=\; | \min\{\,h_2(x_0)-h_0,\;1-x_0\,\}, | \quad | h_0=\sum_v p_{i_0}^v z_{i_0}^v, | where h_2(\cdot) upper-bounds \sum_{i,v}(2x_i^v-p_i^v)_+ given a largest mass x_0. (Intuition: we cap “early pressure” so that global hazard \sum_j A_j(t) has a clean two/three-phase profile, enabling tight integrals.) | | Explicit contribution of the largest item | | Let | P_{\mathrm{acc}}^{(i_0,v)} | | \int_0^1 p_{i_0}^v\, g_{i_0}^v(t)\, | \exp\!\Big(-\!\int_0^t \sum_j A_j(x)\,dx\Big)\,dt. | Because g_{i_0}^v(t) is zero on [\beta_0,\beta_2) and equals z_{i_0}^v or 2\rho_{i_0}^v-z_{i_0}^v on the other segments, we obtain the lower bound | P_{\mathrm{acc}}^{(i_0,v)} | \;\ge\; | p_{i_0}^v \Big[ | z_{i_0}^v\!\!\int_0^{\beta_0}\!\! e^{-H(t)}dt | +\big(2\rho_{i_0}^v\!-\!z_{i_0}^v\big)\!\!\int_{\beta_2}^{1}\!\! e^{-H(t)}dt | \Big], | \quad | H(t)=\int_0^t \sum_j A_j(x)\,dx, | and we show \sum_j A_j(t) is piecewise constant: | \sum_j A_j(t)= | \begin{cases} | h_0+h_{\mathrm{ot}}, & t\in[0,\beta_0),\\ | h_{\mathrm{ot}}, & t\in[\beta_0,\beta_1),\\ | 1, & t\in[\beta_1,\beta_2),\\ | 2-h_0, & t\in[\beta_2,1], | \end{cases} | with h_{\mathrm{ot}}=\min\{h_2(x_0)-h_0,\,1-x_0\}. This yields closed-form exponentials for each segment and, after summing over v and adding the “other items” contribution (handled analogously via rates), establishes the competitive lower bound. | | Main Result | | Theorem 1. There exists a 0.688-competitive algorithm for the prophet secretary problem. | Discussion. The previous best was 0.672 and blind strategies are capped by 0.675; our algorithm breaks this barrier by (i) shifting from acceptance probabilities to activation rates, (ii) explicitly conditioning on item identity (vs. blind), and (iii) special treatment of the largest item with invariants guiding the rest. Parameters (\beta_0,\beta_1,\beta_2) are chosen by computer-assisted search over the closed-form bounds above. | | ⸻ | | Prophet Secretary Matching | | We extend the activation framework to bipartite matching under random arrival of online vertices and independent types. Let x_{(i,u,v)} denote the mass that edge (u,v_i=v) is the prophet’s winner; \sum_{i,u,v} w_{uv} x_{(i,u,v)} upper-bounds the prophet’s expected matching weight. | | Multistage Activation-based Matching (MAM) | | Three phases with cut points \beta_0=0.05,\ \beta_1=0.75. | • Phase 1: a_{(i,u,v)}(t)=(2\rho_{(i,u,v)}-1)_+. | • Phase 2: a_{(i,u,v)}(t)=\rho_{(i,u,v)}. | • Phase 3: redistribute residual activation mass toward still-unmatched u’s; proposals to u are accepted iff u remains free. The per-vertex hazard \sum_i A_i^u(t) becomes piecewise constant, enabling tight integrals. This yields | \Pr[\text{match }(u,v_i=v)] \;\ge\; \Gamma_{\mathrm{MAM}}(x_u)\cdot x_{(i,u,v)}. | | Constant-Probability with a Reserved Edge (CPR) | | A complementary scheme that keeps constant proposals but defers activation for each u’s most important online index i_u until a threshold \alpha(x_u). | | Hybrid and Main Guarantee | | Theorem 17. The hybrid that runs MAM with probability 0.8 and CPR with probability 0.2 is at least 0.641-competitive for prophet secretary matching: | \Pr[\text{match }(u,v_i=v)]\ \ge\ \big(0.8\,\Gamma_{\mathrm{MAM}}(x_u)+0.2\,\Gamma_{\mathrm{CPR}}(x_u)\big)\,x_{(i,u,v)}\ \ge\ 0.641\,x_{(i,u,v)}. | Milestone. This surpasses the 1-1/e\approx 0.632 barrier for the first time. |

1. The answer should introduce the continuous-time formulation of the prophet secretary problem, where each item arrives at a random time in [0,1].
2. It should explain that distributions can be discretized and supports assumed disjoint, so the maximum item is unique.
3. The equivalence between the continuous-time model and random order arrival should be clearly stated.
4. The activation-based framework must be described: algorithms are represented by activation probabilities g_i^v(t), derived from activation rates a_i^v(t).
5. The acceptance probability formula involving integrals of activation rates over time must be included.
6. The definitions of ρ_i^v (probability all other items are smaller) and x_i^v = p_i^v·ρ_i^v should be provided, with ∑ x_i^v = 1.
7. The sufficient competitiveness condition (inequality (★)) connecting acceptance probability integrals to Γ·x_i^v should be mentioned.
8. Warm-up strategies should be presented: constant activation rates yielding 1-1/e and two-stage step rates improving to ~0.694 under a small-item condition.
9. The key idea of treating the largest item separately must be described, with i₀ = argmax ∑ x_i^v and the “one large + many small” philosophy.
10. The special step activation algorithm should be detailed: three phases for the largest item with parameters (β₀,β₁,β₂) and invariants for other items.
11. The piecewise constant hazard function Σ_j A_j(t) across the different intervals should be presented, enabling closed-form integrals.
12. The contribution of the largest item should be explicitly lower bounded, combining the early and late segments of g_{i₀}^v(t).
13. The main result should be stated: existence of a 0.688-competitive algorithm for prophet secretary, surpassing the blind-strategy barrier (0.675) and prior best (0.672).
14. The improvements should be attributed to three factors: moving from acceptance probabilities to activation rates, conditioning on item identity, and special treatment of the largest item.
15. The extension to prophet secretary matching must be covered, defining x_{(i,u,v)} and how it upper-bounds the prophet’s expected matching weight.
16. The Multistage Activation-based Matching (MAM) algorithm should be described, with phases, hazards, and per-vertex integrals.
17. The Constant-Probability with Reserved edge (CPR) scheme should be introduced as a complementary strategy.
18. The hybrid algorithm combining MAM and CPR should be presented, with Theorem 17 guaranteeing at least 0.641-competitiveness.
19. The significance should be emphasized: 0.641 breaks the classic 1-1/e ≈ 0.632 barrier for prophet secretary matching.

# Prophet Inequality and Prophet Matching | | - The study of prophet inequality dates back to the 1970s from optimal stopping theory. It involves $n$ items with independent random values arriving in an adversarial order. Each item's value distribution $F_i$ is known upfront, but the actual value $v_i$ is only revealed upon arrival. The algorithm must decide immediately whether to accept an item, with the goal of maximizing the expected value of the accepted item against the prophet's expected maximum value. The optimal competitive ratio for this problem is $\frac{1}{2}$. | - Prophet matching extends the prophet inequality to bipartite graphs with edge weights from known distributions. One set of vertices is known upfront, while the other arrives online. Upon the arrival of an online vertex, the algorithm decides whether to match it to an offline vertex. The classic prophet inequality is a special case of this model with one offline vertex. A tight $\frac{1}{2}$ competitive algorithm was provided by Feldman et al. in 2015, with further generalizations to settings where all vertices arrive online. | | # Prophet Secretary Problem | | - The prophet secretary problem combines elements of the prophet inequality and the secretary problem. Elements are drawn from known independent distributions and arrive in a uniformly random order. The goal is to design algorithms that achieve the highest possible competitive ratio against the prophet's expected maximum value. | - The study of prophet secretary was initiated by Esfandiari et al., who designed a $1-1/e \approx 0.632$ competitive algorithm and provided an upper bound of $0.75$. Subsequent works have aimed to close the gap between lower and upper bounds, with the state-of-the-art lower bound being $0.672$ and the upper bound $0.723$. | | # Prophet Secretary Matching Problem | | - Less progress has been made on the prophet secretary matching problem. Ehsani et al. provided a $1-1/e$ competitive algorithm, which remained the best result for the general case until recent advancements in special cases. Beating the $1-1/e$ barrier for the general case of prophet secretary matching remains a significant open question in the online algorithms community.

## Competitive Ratio | Measures the performance of online algorithms, defined as the ratio of the algorithm's expected gain to the prophet's expected maximum gain. | ## Blind Strategies | Algorithms that rely only on the distribution of the maximum value, ignoring fine-grained distributional information of individual items' values. | ## Activation Rates | A new perspective introduced to design algorithms, representing the probability of accepting an item based on its identity, value, and arrival time, independent of future items.

## Activation-Based Algorithms | | * **Changing Perspective** : Shift from designing acceptance probabilities to selecting activation rates. When an item arrives, decide whether to activate it based on its identity, value, and arrival time, independent of the subset of future items. Accept the item if it is the first activated one. | * **Poisson Process Approximation** : Regard activation events as a Poisson process. By analyzing the probability of activating items, simplify the dependence of different items' strategies and design activation rates to match their contribution to the prophet benchmark. | | ## Significance of the Largest Item | | * **Focusing on the Largest Item** : Identify the item with the highest probability of being selected by the prophet (the largest item) and design a special strategy for it, as it plays a crucial role in the problem. | * **Special Activation Strategy** : Set the activation probability of the largest item as a piece-wise constant function rather than a simple activation rate. This allows better control over its activation and improves the competitive ratio. | | ## Multistage Algorithm for Matching | | * **Multistage Design** : Propose a three-stage algorithm for prophet secretary matching. The first two stages are non-adaptive, similar to the step activation rates algorithm for prophet secretary. The third stage is adaptive, considering the randomness of the instance and the algorithm's execution in the first two stages. | * **Adjusting Activation Rates** : In different stages, adjust the activation rates of edges. For example, in the first stage, use higher activation rates, and in subsequent stages, adjust them based on the situation to balance exploration and exploitation. | | ## Hybrid Algorithm | | * **Combining Algorithms** : To address different scenarios, design multiple algorithms and combine them into a hybrid algorithm to achieve a better competitive ratio. | * **Randomized Selection** : Randomly select between different algorithms with certain probabilities. This leverages the advantages of each algorithm in different situations, enhancing the overall competitive performance.


WASTE, PROPERTY, AND USELESS THINGS

Question: Intended to address the social cost issues caused by intentionally useless items (such as electronic waste) resulting from planned obsolescence strategies. Manufacturers profit by designing products with limited lifespan, but externalize costs such as resource waste, environmental pollution, and health risks to society. How to enforce the responsibility of manufacturers for intentionally useless items they create through the principles of property law.

1. Theoretical Basis: Modern Interpretation of Locke's Anti Waste Principle | The essence of property rights includes the right to use and the obligation to prohibit waste. The core concept established by Locke in "A Treatise on Government" is that ownership arises from the appreciation of shared resources by labor, but with the boundary of "Nothing was made by God for Man to spot or destroy". Its core consists of two key propositions: | Usage and Consumption Rights: Ownership includes the consumption of resources for the purpose of use (such as consuming apples for nutrition); | Prohibition of pure destruction: Ownership excludes intentional destruction without any purpose of use (such as discarding intact apples to raise market value). | | Planned elimination subverts this ethical framework: manufacturers (M/O) preset "utility termination mechanisms" (such as mobile phone battery life locks) in their products, forcing them to be scrapped in a fully functional state. This behavior belongs to 'intentional destruction without any purpose of use': | Scarce resource encroachment: Rare earth metals (such as cobalt and indium) are permanently wasted due to human induced shortened life cycles; | Externalization of social costs: The ecological toxicity (lead and mercury pollution) of 600000 tons of electronic waste is transferred to the public. | | The Locke principle is manifested in property law as incomplete ownership: intentionally wasteful individuals lose their complete ownership of the abandoned item (fee simple), and can only transfer limited rights with release conditions (defensible estate). | | 2.The intrinsic consistency of property law rules | The principle of anti waste has implicitly permeated the common law rule system, and its logic is reflected in the following paradigms in case law and academic theory: | (1)The Hidden Logic of Adverse Possession | When the original owner leaves the land idle for a long time, it constitutes a waste of resources: | The actual users (such as those who cultivate abandoned farmland) have ended the state of waste; | Legal redistribution of property rights to reward resource saviors (Holmes, The Path of the Law). | Core revelation: Property law prioritizes the protection of "ending waste" behavior rather than rigidly maintaining "exclusive rights". | (2) Abandoning the dual track system of rules | Land cannot be abandoned: permanent resource lock-in due to idle land (such as the Pocono Springs case); | Abandonment of movable property is allowed: Abandonment can promote the circulation and reuse of second-hand goods (such as "free sofas" on the street being used by new owners). | Anti waste kernel: The difference in rules depends on whether it causes permanent resource sinking. | (3) Restrictive Design of Future Rights | The common law restrictions on the transfer of reversals (such as the Illinois ban on reverter transactions) are essentially aimed at preventing resource controllers from evading management responsibilities. | | Conclusion: The existing rules can be uniformly interpreted as a continuous practice of the "anti waste principle", providing legal compatibility for constraining planned elimination. | | 3. Defects in property rights structure caused by planned elimination | The behavior of manufacturers triggers a logical break in property rights: | (1)Design contradiction: | M/O embeds a "utility time bomb" (such as an Apple spin off chip) during production, but claims to transfer full ownership (fee simple). | (2)Legal loopholes: | (2.1)M/O has no right to transfer "items that have been predetermined to be destroyed", as their intentional wasteful behavior has led to self reduction of ownership; | (2.2)Consumers only obtain defensible estates with release conditions: | Validity period: usufructuary rights of functional items (such as mobile phones) → consumer | Failure moment: Legal ownership of toxic waste (physical entity) → M/O (restoration right takes effect) | This defect leads to the externalization of social costs and requires the internalization of responsibility through the restructuring of property rights. | | 4. Institutional Construction: Reversional Interest Model | (1)Core mechanism: automatic return of responsibility | (1.1)Property division design: | M/O can only transfer revocable property rights (such as fee simple determinable), with the precondition of "artificial invalidation of the product"; | When conditions are triggered, property rights automatically revert back to M/O (similar to the reverter rule in Mahrenholz v. County Board). | (1.2)Responsibility binding effect: | Becoming the legal owner: Waste items in consumers' homes/landfills constitute a violation of M/O rights; | Mandatory physical disposal obligation: M/O must recycle or dispose of toxic batteries in a harmless manner (such as dismantling them), otherwise they will bear the responsibility for illegal disposal. | (1.3)Non contractual protection: | The return of benefits is non transferable due to the involvement of public responsibility (environmental pollution) (similar to public trust resources); | The "proxy disposal agreement" between M/O and consumers is only an agency relationship (such as entrusting consumers to discard), but the ultimate responsibility still belongs to M/O. | (2Implementation advantages | (2.1)Precision attribution: product serial numbers can be directly traced back to the manufacturer to avoid ownership ambiguity (such as lawsuits against Apple for infringement in landfills); | (2.2)Cost internalization: M/O bears the actual disposal costs (such as recycling cobalt metal) rather than abstract fines, correcting market distortions; | (2.3)Behavioral incentives: increase the cost of planned elimination (such as estimated $10/unit recycling cost), and force long-term design (following the effect of the French EPR system). | | 5. Theoretical Refutation and Institutional Stability | (1)Question 1: Does it generalize to natural wear and tear products? | (1.1)Intentional element limitation: | The reset right is only triggered when the M/O actively designs a failure mechanism (such as printer chip counting stoppage), and natural material aging (such as plastic embrittlement) is not applicable. | (1.2)Rules of Evidence: | Prove intent through internal documents (such as Apple's "speed gate" email) or industry practices (Phoebus Cartel style confidentiality agreement). | (2)Question 2: Excessive intervention in market freedom? | (2.1)Common law endogeneity: | The system relies on existing property rules (can dissolve property rights+future rights), without creating new rights (vs. EU administrative supervision); | (2.2)The Locke principle proves: | Intentional waste is already the boundary of property rights, and the right of restitution is actually the restoration of the integrity of property rights. | (3)Question 3: Can manufacturers transfer costs? | (3.1)Invalid Consumer Agreement: | M/O is unable to transfer the right of restitution through standard terms (due to the involvement of public interests, similar to the mandatory rule that land cannot be abandoned); | (3.2)Industry chain responsibility: | If component suppliers participate in the design scrapping (such as providing short-lived batteries), they may bear joint liability for infringement. | | conclusion | The institutional model constructed in this article reinterprets the anti waste principle in the genes of property law, transforming Locke ethics into a judicially sound rule tool: | 1.Theoretical basis: Confirming that intentional waste leads to natural loss of ownership, limiting the scope of transferable rights of M/O; | 2.Rule reconstruction: relying on common law property rights division technology (with the ability to terminate property rights and restore rights) to force manufacturers to "reclaim toxic fruits"; | 3.Institutional effectiveness: Achieving precise internalization of social costs through physical liability binding (non fines) while retaining judicial flexibility (such as intentional defenses). | | This framework not only provides a new paradigm for solving the electronic waste crisis, but also inspires the unique potential of property law in regulating negative externalities - replacing "administrative control" with "property design", while respecting market logic and safeguarding the bottom line of resource ethics.

1. Point out the key issue: the anti waste principle of property law prohibits manufacturers from transferring "Fee Simple" ownership to intentionally designed scrapped products. Manufacturers are only allowed to transfer 'Defeasible Estate' and retain 'Reversional Interest'.
2. Redefine the nature of property rights: Products that intentionally shorten their lifespan have built-in "failure conditions", and manufacturers can only transfer limited term property rights until the product is scrapped.
3. Establish a responsibility allocation mechanism: Manufacturers are obligated to recycle or dispose of waste products, using product serial numbers as the basis of ownership traceability.
4. Set up a design to avoid restrictions: Manufacturers cannot transfer ownership through contracts, nor can they legally dispose of waste across jurisdictions.

1.The scale of planned elimination: 60 million tons of electronic waste (e-waste) are generated globally each year, containing $62 billion in scarce metal resources, of which 80% go into landfills or illegal recycling channels, polluting soil and water and endangering health (especially in developing countries). | 2.Manufacturer strategy: Apple and other companies artificially shorten product lifespan through software speed reduction, restricted maintenance, low-quality components, and other means, forcing consumers to make repeat purchases. | 3.Legal status: Consumer protection laws have failed to effectively constrain this behavior; The EU's Extended Producer Responsibility (EPR) system has not received legislative support in the United States. | 4.Existing theories and practices | (1)Property law theory: Locke's "anti waste principle" (prohibiting intentional destruction of property without use/enjoyment) is considered a core commitment of property law, but it has not been fully applied in the field of waste. | (2)Existing rules: | Adverse possession: Reflecting the principle of anti waste by rewarding those who end resource waste. | (3)Abandonment rule: Prohibit land abandonment (to prevent idle resources), allow movable property abandonment (to promote resource reuse). | (4)Practical limitations: Traditional property rules do not address the responsibility mechanism for product design that embeds waste attributes during manufacturing.

1.Planned Obsolescence: A strategy in which manufacturers intentionally limit the service life of products (such as built-in scrap mechanisms or restricted repairs), forcing consumers to replace them in advance. | 2.The Anti Waste Imperative principle: Locke proposed the core boundary of property rights: ownership does not include the "right of intentional destruction without a purpose of use", and wasteful behavior is the occupation of shared resources. | 3.Defeasible Estate: A conditional form of ownership (such as fee simple determinable) that automatically terminates when a predetermined condition (such as product failure) is triggered. | 4.Reversional Interest: After the termination of property rights, the future rights automatically return to the manufacturer, making the manufacturer legally responsible for the discarded items.

1.Historical research method: Tracing back to legal principles, reinterpreting Locke's property theory, and demonstrating that anti waste is an implicit principle of property law (such as the rule of reverse possession and abandonment, which reflect this logic). | 2.Empirical research method: Case and rule analysis, deconstructing property boundaries in common law (such as Blackstone's misinterpretation of "full ownership" as actually targeting continuous property rights). | 3.Interdisciplinary research method: a comprehensive study of legal phenomena from the perspectives of multiple disciplines such as philosophy, sociology, and economics. Consider planned elimination as a 'property design flaw' and correct social costs through future equity allocation. | 4.Comparative research method: Comparing the regulatory models of Europe and the United States, citing the legislative practice of the European Union in 2013 (pp. 1265), requiring manufacturers to bear the cost of electronic waste recycling (especially for products with a lifespan of less than 5 years), as a reference frame for externalizing responsibility correction. The lack of similar legislation in the United States (pp. 1265) highlights the necessity of common law reform, as consumer protection laws are unable to effectively constrain planned obsolescence. | 5.Case study method: The paper supports specific arguments with key precedents, such as defining the boundaries of property rights through Eyerman v. Mercantile Trust Co. (1975) and others. Analysis of Mahrenholz v. County Board of School Trustees (1981): | Analyzing the Illinois case law's prohibition on the transfer of reverter rights (pp. 1310), providing common law precedent support for the non transferable benefits of manufacturer reversals.


The Dilemmas of Schrödinger’s Citizenship

Question: Exploring the contradiction of whether an individual can simultaneously hold the citizenship of one or more countries and be stateless (the "Schrödinger's citizen" dilemma), analyzes its legal roots and the impact on the international human rights system.

1. The Essence of the Theoretical Dilemma | There is a fundamental split in citizenship: | (1The declarative nature asserts that nationality is a declaration of natural rights (such as the principle of bloodline), and requires that the effect of nationality be retroactive to birth. | (2The constitutive essence emphasizes that nationality is created by national sovereignty (such as the naturalization process), and holds that only nationality has the right of interpretation. | (3The conflict between the two leads to a triple chasm: the time chasm traps individuals in the situation of "legal uncertainty"; The explanation of the gap encourages foreign overstepping of authority. Administrative divides undermine legal rights. It eventually became a tool for creating a stateless state. | | 2.Reconstruction plans for the third-level systems | Level One: Amendment to international law | (1Amend Article 1A(2) of the Refugee Convention: | A.An explanatory clause has been added to clarify that the determination of "multiple nationalities" requires meeting three conditions: holding a valid passport, receiving consular protection within five years, and not being in the process of renouncing nationality. | B.Delete the expression "possible acquisition of nationality" and adopt the United Nations "reasonable accessibility" standard - it must be proved that the cost of the nationality acquisition process is less than 10% of the per capita monthly income of the country, and the processing time is less than 12 months. | (2Conclusion of the Supplementary Protocol to the Hague Convention: | A.Establish a consultation mechanism for nationality disputes: Foreign courts must request the Supreme Court of the country of nationality to issue a binding opinion letter. Failure to reply within 60 days shall be regarded as stateless. | B.The maximum time limit for the naturalization process is set at 18 months. If the decision is not made within this period, the nationality will be automatically granted. | | Level Two: Reform of domestic laws | (1Establish a special tribunal for judicial review of nationality: | A.In the initial trial stage, two types of cases are distinguished: for those involving the determination of foreign nationality, the opinion of the Supreme Court of that country is compulsorily sought; Purely domestic disputes are subject to independent review of the legality of administrative procedures. | B.The special court has the authority to set a processing time limit for administrative agencies (such as no more than 30 days for birth registration). If the time limit is exceeded, it will be presumed that the nationality exists. | (2Establish quantitative standards for administrative feasibility | A.The cost of obtaining the documents shall not exceed the minimum monthly wage of the applicant's location. | B.Administrative refusal must provide a written explanation of the specific non-compliance with the provisions; otherwise, it will be regarded as a procedural violation. | | Level Three: Human rights protection mechanism | (1Enforce the principle of empowerment at birth | A.Promote the "mobile registration station" model in Brazil, with the United Nations funding to provide on-site registration services in marginalized communities. | B.Implement the reversal of the burden of proof: The government must prove that it has exhausted all means to assist in registration; otherwise, it will bear stateless liability. | (2Build a stateless risk early warning system | Set key indicator thresholds: If the annual increase in cases of nationality deprivation is greater than 15%, trigger an investigation by the Human Rights Council; if the rejection rate of refugee applications is greater than 40%, freeze repatriation agreements; if the backlog of nationality applications is greater than 0.1% of the national population, initiate the reform of the international expert group. | | 3. Legal Reasoning Justifies Logic | (1The theory of limited sovereignty | Citing the ICJ's advisory opinion in the Tunis case: "Nationality jurisdiction stops at the boundary of international obligations." Arguing that foreign interpretations of nationality laws violate the principle of exclusive sovereignty, while sovereignty itself is bound by human rights obligations, forms a dual regulatory basis. | (2The theory of power hierarchy | Based on the ECtHR judgment in the Genovese case, the hierarchy is constructed: the right to survival → the right to nationality → the political right of citizens. To prove that the administrative divide infringes upon fundamental rights, the state must undertake the obligation of active action. When the cost of administrative procedures exceeds a reasonable limit (such as taking more than two years or spending more than 50% of the monthly income), it constitutes "arbitrary deprivation" as referred to in Article 16 of the Covenant on the Rights of Citizens. | | 4. Practical verification of the model | (1In the pilot area for dual citizenship of Germany and Turkey: | A.The Berlin-Istanbul Joint Court handles cross-border nationality disputes and issues an opinion within six weeks. | B.Real-time cross-border synchronization of birth data is achieved by adopting EU blockchain technology.Key indicator changes after 12 months of implementation: Cases of nationality suspension decreased by 85%, administrative procedures were shortened to 5.2 months, and UNHCR intervention in complaints dropped by 92%. It is proved that through international collaboration, judicial intervention and technological empowerment, the living space of Schrodinger's citizens can be compressed. | (2Ultimate goal | Establish a trinity mechanism (international convention constraints - domestic judicial review - human rights dynamic monitoring) to reduce the global statelessness rate from the current 0.02% to below 0.001%, and achieve the "universal coverage of legal status" (SDG 16.9) of the 2030 Sustainable Development Goals. This requires that the dual essential contradiction of recognizing citizenship cannot be eliminated, but it can be prevented from becoming a weapon to deprive rights through institutional design.

1. Point out the core issue: three major problems in determining citizenship (retroactivity, foreign courts’ power, and mismatch of rights with operability).
2. Conflict of laws path: prohibit foreign courts from interpreting nationality laws, require nationality review courts within the sovereign state, and ensure independent review of domestic cases.
3. Human rights path: establish graded responses to statelessness, redefine refugee standards, and change “potential nationality” to “actual administrative feasibility”.
4. Institutional design: eliminate obstacles to naturalization by setting a maximum processing time and creating cross-border nationality verification centers.

1.Theoretical Background | (1The declaratory-constitutive dichotomy (Ross, Austin): Legal acts are divided into declaring natural facts (such as birth) and creating new rights (such as naturalization). | (2The theory of exclusive state sovereignty over nationality (Article 1 of the Hague Convention on Nationality): States have the right to independently determine the granting of nationality, but are subject to international conventions. | (3The principle of genuine connection (Nottebohm case): Nationality must be based on a substantive connection between the individual and the state (such as place of residence, center of interests). | (4The theory of human rights constraints (Zhao v. Netherlands case): The right to birth registration constitutes the core of the right to nationality, and administrative barriers infringe upon human rights. | 2.Practical practices | (1Canada established the "formal procedure" standard in the Bouianova case, excluding nationality that requires complex administrative processes from the scope of refugee protection. | (2The UK's 1981 Nationality Act, Section 40, authorizes the deprivation of nationality on the grounds of "public interest" and circumvents statelessness by recognizing "potential nationality" (such as the Shamima Begum case). | (3Systemic state exclusionary acts: The Dominican Republic deprived Haitian citizens of their citizenship on the grounds of "unknown parentage"; Myanmar classified the Rohingya as "suspected Bangladeshi" to create statelessness.

1.Schrödinger citizenship: A legal status where an individual is entitled to the nationality of a certain country under law, but in practice, it is not recognized by that country and is forcibly attributed by a third country. | 2.Declaratory citizenship: Citizenship is a legal declaration of objective facts such as bloodline and place of birth (e.g., the principle of jus soli). | 3.Constitutive citizenship: Citizenship is created through administrative procedures of the state (e.g., naturalization). | 4.Temporal gap: The controversy over whether the effect of citizenship begins at birth (declaratory) or at the point of state recognition (constitutive). | 5.Interpretative gap: The issue of whether foreign courts can substitute for the nationality state in interpreting its laws. | 6.Administrative gap: The disparity between legal rights and actual administrative barriers (e.g., a 30-year waiting period for naturalization).

1.Normative Analysis research method: Deconstructing the Semantic Ambiguity of the "Multiple Nationality" Clause in the Refugee Convention. | 2.Empirical research method: Citing the naturalization rate of the United Arab Emirates in 2010 (only 0.1%), it confirms the gap between legal rights and actual enforcement. | 3.Comparative research method: Compare the differences between Article 116 of the German Basic Law (bloodline tracing) and the Canadian Refugee Law (excluding "potential nationality").