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potts The Ising Hamiltonian has been previously realised in optical networks, most notably in the degenerate optical parametric oscillators (DOPOs) dopo , where the phase projection to [MATH] and [MATH] is due to the second order nonlinear crystal placed in an optical cavity. However, only resonant forcing is present t... |
To simulate the behaviour of the gain-dissipative system one can use Eqs. ( ); however, since simplifying assumptions were made to arrive at Eq. ( ) it is important to verify that a real system shows the transition from achieving the minimum of the continuous (XY) model to the minimum of the discrete model when the res... |
[EQUATION] where [MATH] is the condensate wavefunction, [MATH] is the density profile of the hot exciton reservoir, [MATH] is the polariton effective mass, [MATH] and [MATH] are the strengths of the effective polariton-polariton interaction and the blue-shift due to interactions with non-condensed particles, respective... |
To illustrate how the polariton lattice minimizes the discrete Ising or Potts models with or without external fields we study the behaviour of the unit polariton lattice cells when they are subjected to the effect of the resonant forcing. In what follows we explore the behaviour of the system without density and coupli... |
To illustrate the transition from solving the XY model to the Potts model we consider four condensates arranged at the corners of a square with antiferromagnetic coupling between the nearest neighbours and ferromagnetic coupling along the diagonal. Figure c shows the solution of Eqs. ( ) without the resonant terms ( [M... |
Finally, we combine two resonant forcing terms: the resonance [MATH] and either resonance [MATH] or [MATH] in Eq. ( ) to combine the effect of an external ”magnetic” field in the Ising or 3-state Potts models respectively. We take [MATH] and [MATH] . Such external field penalises the objection function if the phases of... |
In conclusion, we formulated an implementation of the discrete spin Hamiltonians such as the Ising and Potts models with or without the external fields in a lattice of non-equilibrium Bose-Einstein condensates or coupled lasers. We show that if a resonant pumping is employed together with a non-resonant excitation then... |
Acknowledgements The authors acknowledge financial support from the NGP MIT-Skoltech. K.P.K. acknowledges the financial support from Cambridge Trust and EPSRC. |
# Source: arxiv 1807.00317 # Title: An Improved Envy-Free Cake Cutting Protocol for Four Agents # Sections: all # Downloaded: 2026-03-02T09:23:12.140602+00:00 |
An Improved Envy-Free Cake Cutting Protocol for Four Agents Abstract We consider the classic cake-cutting problem of producing envy-free allocations, restricted to the case of four agents. The problem asks for a partition of the cake to four agents, so that every agent finds her piece at least as valuable as every othe... |
. The main drawback of these new algorithms, however, is that they are quite complicated and with a very high query complexity. With four agents, the number of queries required is close to 600. In this work we provide an improved algorithm for four agents, which reduces the current complexity by a factor of 3.4. Our al... |
by incorporating new insights and simplifying several steps. Overall, this yields an easier to grasp procedure with lower complexity. |
Introduction Producing an envy-free allocation of an infinitely divisible resource is a classic problem in fair division. As it is customary in the literature, the resource is represented by the interval [MATH] , and each agent has a probability measure encoding her preferences over subsets of [MATH] The goal is to div... |
The problem has a long and intriguing history. It has been long known that envy-free allocations exist for any number of agents, via non-constructive proofs |
For algorithms, the standard approach is to assume access to the valuation functions via evaluation and cut queries (see Section |
). Under this model, we are interested in counting the number of queries needed for producing an envy-free allocation. For two agents, the famous cut-and-choose protocol requires only two queries. For three agents, the procedure of Selfridge and Conway |
guarantees an envy-free allocation after at most 14 queries. For four agents and onwards, however, the picture changes drastically. The first finite, yet unbounded, algorithm was proposed by |
. This was followed up by other more intuitive algorithms, which are also unbounded, e.g., Finding a bounded algorithm was open for decades and positive results had been known only for certain special cases, like piece-wise uniform or polynomial valuations |
It was only recently that a major breakthrough was achieved by Aziz and Mackenzie, presenting the first bounded algorithms, initially for four agents |
, and later for an arbitrary number of agents Despite these significant advances, the algorithms of are still of very high complexity. For an arbitrary number of agents, [MATH] , the currently known upper bound involves a tower of exponents of [MATH] , and even for the case of four agents, the known algorithm requires ... |
Contribution: We focus on the case of four agents and present an improved algorithm that reduces the query complexity roughly by a factor of [MATH] (requiring [MATH] cut queries and [MATH] evaluation queries). Our algorithm utilizes building blocks that are similar to the ones used by |
, but by incorporating new insights and simplifying several steps, we obtain a solution with significantly fewer queries. The main differences between our work and |
are highlighted at the end of this section. Our algorithm works by maintaining a partial allocation along with a leftover residue. Throughout its execution, it keeps updating the allocation and reducing the residue, until certain structural properties are satisfied. These properties involve the notion of domination , w... |
Further related work: We refer the reader to the book chapters for a more proper treatment of the related literature. Towards simplifying the algorithm of Aziz and Mackenzie |
, the work of Segal-Halevi et al. (see their Appendix B) proposes a conceptually simpler framework, without, however, improving the query complexity. Apart from the algorithmic results mentioned above, there has also been a line of work on lower bounds. For envy-freeness, Stromquist |
showed that there is no finite protocol for producing envy-free allocations where all the pieces are contiguous. Later on, Procaccia |
established an [MATH] lower bound for producing non-contiguous envy-free allocations. Apparently, there is still a huge gap between the known lower and upper bounds for the problem for any [MATH] Interestingly, for the stricter notion of strong envy-freeness , where we require each agent to believe she is strictly bett... |
An Overview of the Algorithm We start with a high level description of the main ideas. As with most other algorithms, our algorithm maintains throughout its execution a partial allocation of the cake, along with an unallocated residue. The goal is to keep updating the allocation and diminishing the residue, with the in... |
Phase One. We find this first phase of particular importance, as it is also the most computationally demanding one. Here the goal is to get a partial envy-free allocation in which some agent is dominated by two other agents as in Figure 1(a) In order to establish dominations among agents, we use as a subroutine the so-... |
Phase Two. Suppose that at the end of phase one, agent [MATH] is dominated by agents [MATH] and [MATH] The goal in the second phase is to produce a partial envy-free allocation where both [MATH] and [MATH] dominate both [MATH] and [MATH] To achieve this goal, we execute Core twice on the residue with [MATH] as the cutt... |
Phase Three. Since both [MATH] and [MATH] are now dominated by [MATH] and [MATH] , we can simply execute the cut-and-choose protocol for [MATH] and [MATH] on the remaining residue. |
Similarities and Differences with the Aziz-Mackenzie Algorithm in Our algorithm uses similar building blocks as the algorithm for four agents in |
, combined with new insights. Namely, our Core and Correction protocols on a high level serve the same purpose as the core and the permutation protocols in |
. Conceptually, a crucial difference is the target structure of the domination graph. The initial (and most query-demanding) step of |
is to have every agent dominate two other agents. Here, our goal in phase one is to have just one agent dominated by two other agents . Once this is accomplished, it is possible to reach a complete envy-free allocation much faster. Another important difference is the implementation of the Core protocol itself. Our vers... |
, such as the current allocation, and it is not required to always output a partial envy-free allocation of the current residue. This extra flexibility allows us to avoid the tedious case analysis stated in the core protocol of |
and, at the same time, further reduce the number of queries. Preliminaries Let [MATH] be a set of four agents. The cake is represented as the interval [MATH] ; a piece of the cake can be any finite union of disjoint intervals. Each agent [MATH] is associated with a valuation function [MATH] defined on all finite unions... |
Normalization: [MATH] Additivity: for all disjoint [MATH] [MATH] Divisibility: for every [MATH] and every [MATH] , there exists [MATH] such that [MATH] Note that this implies that [MATH] , for all [MATH] |
Nonnegativity: for every [MATH] it holds that [MATH] By [MATH] we denote the allocation where agent [MATH] is given the piece [MATH] |
Definition 1 (Envy-freeness) An allocation [MATH] is envy-free , if [MATH] , for all [MATH] , i.e., every agent prefers her piece to any other agent’s piece. |
We say that [MATH] is a partial allocation, if there is some cake that has not been allocated yet, i.e., [MATH] . The unallocated cake is called the residue During the execution of the algorithm the residue diminishes, until eventually it becomes the empty set. As we noted, an important notion is that of domination or ... |
. It will be insightful to think of a graph-theoretic representation of our goals, via the domination graph of the current allocation. |
Definition 2 (Domination and Domination Graph) Given a partial allocation [MATH] and a residue [MATH] , we say that an agent [MATH] |
dominates another agent [MATH] , if [MATH] . That is, [MATH] would not be envious of [MATH] even if [MATH] were allocated all of [MATH] . The domination graph with respect to [MATH] is a directed graph where the nodes correspond to the agents and there exists a directed edge [MATH] if and only if agent [MATH] dominates... |
Achieving certain patterns in the domination graph can make the allocation of the remaining residue straightforward. For example, if there exists a node [MATH] with in-degree [MATH] , allocating all of the residue to agent [MATH] results in an envy-free allocation. As another example, the protocol of |
tries to get a domination graph where every node has out-degree at least [MATH] . In our algorithm, we also enforce a certain structure on the domination graph. |
The Robertson-Webb Model The standard model in which we measure the complexity of cake cutting algorithms is the one suggested by Robertson and Webb |
and formalized by Woeginger and Sgall In this model, two kinds of queries are allowed: Cut queries: given an agent [MATH] , a point [MATH] and a value [MATH] , with [MATH] the query returns the smallest [MATH] such that [MATH] |
Evaluation queries: given an agent [MATH] and an interval [MATH] , return [MATH] Virtually all known discrete cake-cutting protocols can be analyzed within this framework. For example, the cut-and-choose protocol is implemented as follows: the algorithm makes one cut query for agent 1 with [MATH] , starting from [MATH]... |
Conventions on Ties, Marks, Partial Pieces, and Residues All algorithms in this work ignore ties. However, assuming an appropriate tie-breaking scheme, this is without loss of generality (also see the discussion in |
). We follow some conventions—also adopted in related work—when it comes to handling trims and partial pieces. In various steps during the algorithm, one agent cuts the residue into pieces, and the other agents are asked to place marks on certain pieces. We always assume that marks are placed starting from the left end... |
Definition 3 Given a partition of the residue into four pieces, we say that an agent performs an [MATH] -mark, if she places a mark on each of her [MATH] most valuable pieces so that the resulting partial pieces all have the same value as her [MATH] -th favorite piece. |
In the description of the algorithm we use [MATH] -marks and [MATH] -marks. Of course, after all marks are placed, each connected piece may have multiple marks on it. Whenever a connected piece [MATH] is only partially allocated, the part [MATH] of [MATH] that is allocated is always the interval between the second righ... |
beyond (resp. strictly beyond ) a mark [MATH] , if [MATH] is allocated [MATH] with [MATH] (resp. [MATH] ). Note that in the above discussion the residue is seen as a single interval, while in fact it may be a finite union of intervals. We keep this view throughout this work as it is conceptually easier and allows for a... |
The Algorithm The main result of our work is the following. Theorem 1 The Main Protocol returns an envy-free allocation and makes at most [MATH] cut queries, and [MATH] evaluation queries. |
We discuss first the main steps of our algorithm and provide the relevant definitions and key properties, needed for the proof of correctness, in Section 3.1 |
Phase One. This is the most important part of the protocol, and computationally the most demanding one. The goal in phase one is to get a partial envy-free allocation, where some agent is dominated by two other agents, i.e., the underlying domination graph has a node with in-degree at least [MATH] , as depicted in Figu... |
Core Property 1 The cutter and at least one more agent receive complete pieces, each worth exactly [MATH] of the value of the current residue according to the cutter’s valuation. |
Core Property 2 The allocation output by any single execution of Core when no agent is excluded from competition, is a (possibly partial) envy-free allocation. |
The above properties allow us to deduce an important fact: if Core is executed at least twice with the same agent as the cutter, then this cutter dominates at least one agent in the resulting allocation. In fact, we can be more specific about the agent who gets dominated. The important observation here, stated in Lemma... |
Definition 4 Let [MATH] be an allocation produced by a single run of Core . Among the four pieces given to the agents, the partial piece that is least desirable to the cutter is called the insignificant piece of [MATH] |
Hence, if we run Core twice, say with agent [MATH] as the cutter, we enforce one edge in the domination graph. In order to proceed further and obtain a node with in-degree two, we first attempt, as an intermediate step, to have a domination graph where one node has out-degree equal to two. One may think that the interm... |
any number of times, there is no guarantee that new dominations will appear; the same agent may receive the insignificant piece in every iteration. |
To fix this issue, it suffices to run Core 4 times with agent [MATH] as the cutter and then make a small correction to one of the [MATH] partial allocations produced by Core In particular, denote by [MATH] , with [MATH] , the suballocation output by the [MATH] th execution of Core within the for loop of line of Main Pr... |
Definition 5 (Gain) Let [MATH] be the current partial allocation of the cake, and [MATH] be a suballocation of [MATH] , i.e., [MATH] for [MATH] . Further, let [MATH] be the set of agents that are dominated by [MATH] in [MATH] and [MATH] . Then the gain of [MATH] with respect to [MATH] |
[MATH] , is the difference between [MATH] and the maximum value of [MATH] for a piece in [MATH] given to any agent in [MATH] i.e., [MATH] |
Phase One for [MATH] to [MATH] do Run Core on the current residue with agent [MATH] as the cutter. if the same agent got the insignificant piece in all 4 executions of Core |
then Find [MATH] such that [MATH] for all [MATH] Run Correction on [MATH] Run Core on the residue with agent [MATH] as the cutter. |
if there is some agent [MATH] not dominated by agent [MATH] then Run Core on the residue with agent [MATH] as the cutter, excluding agent [MATH] from competition. |
else 10 Run the Selfridge-Conway Protocol on the residue for agents [MATH] [MATH] , and [MATH] , and terminate. Now, if the algorithm has not terminated, some agent [MATH] is dominated by two other agents [MATH] and [MATH] . Let [MATH] be the remaining agent. |
Phase Two 11 for [MATH] to [MATH] do 12 Run Core on the current residue with agent [MATH] as the cutter, excluding from competition any one from [MATH] who dominates two non-cutters. |
13 if [MATH] and [MATH] are not both dominated by [MATH] and [MATH] then 14 Let [MATH] be the agent who got the insignificant piece in the last two calls of Core |
15 Run Correction on the suballocation (out of the last two) where [MATH] was smaller. At this point both [MATH] and [MATH] dominate both [MATH] and [MATH] |
Phase Three 16 Run Cut and Choose on the current residue for agents [MATH] and [MATH] Algorithm 1 Main Protocol [MATH] Using Definition , we identify a suballocation among [MATH] , where the gain of each agent is small compared to her combined gain from the other three suballocations (line of the algorithm). The existe... |
Correction Property 1 The insignificant piece of [MATH] is given to a different agent in [MATH] . In particular, it is given to an agent that has marked it in [MATH] |
Correction Property 2 If a non-cutter was allocated her favorite unmarked piece in [MATH] , she will again be allocated a piece of the same value in [MATH] |
Assume there is no agent dominating everyone else, meaning that [MATH] is defined for all [MATH] . For a partial envy-free allocation like [MATH] , the gain of any agent is nonnegative. However, this may not be true for [MATH] , as it is not necessarily envy-free. What we need is for [MATH] to be envy-free, and towards... |
Correction Property 3 [MATH] for all agents [MATH] By Correction property , the insignificant piece has changed hands after line . This allows us to make one extra call to Core in order to enforce one more domination (line ). Hence, the intermediate step is completed and we know that agent [MATH] dominates at least 2 o... |
At this point there are various ways to proceed, each with a different query complexity. E.g., we could repeat the whole process so far, but with agents [MATH] and [MATH] as cutters, and get at least 6 edges in the domination graph. This would ensure a node with in-degree two, but it requires several calls to Core . In... |
Remark 1 The intermediate step of getting a node with out-degree two has also been utilized in . The goal there however was to make every agent dominate two other agents, whereas we only needed this to hold for one agent. |
Phase Two. Suppose phase two starts with a partial envy-free allocation where some agent, say [MATH] , is dominated by agents [MATH] and [MATH] (Figure 1(a) ). Our next goal is to produce a partial envy-free allocation where both [MATH] and [MATH] dominate both [MATH] and [MATH] (Figure 1(b) ). To achieve this goal, we... |
Core Property 3 Assume we run Core with [MATH] as the cutter, and suppose agent [MATH] is dominated by the other two non-cutters, [MATH] and [MATH] , neither of whom dominates the other. Then, (1) |
[MATH] gets her favorite of the four complete pieces without making any marks, (2) at least three complete pieces are allocated, and (3) if a non-cutter, say [MATH] , gets a partial piece, then the remaining non-cutter, [MATH] , is indifferent between her piece and [MATH] ’s piece. |
Using this property, we can show that after one call to Core (1st execution of line ), agents [MATH] and [MATH] will both dominate either [MATH] |
or [MATH] However, we need domination over both [MATH] and [MATH] . The second call to Core (2nd execution of line ) ensures that we can again resort to the Correction protocol. If, after the two calls to Core , only one of [MATH] and [MATH] , say [MATH] , is dominated by both [MATH] and [MATH] , then running Correctio... |
Phase Three. Since both agents [MATH] and [MATH] are dominated by [MATH] and [MATH] , we just execute the cut-and-choose protocol for [MATH] and [MATH] , where [MATH] cuts the residue in two equal pieces and [MATH] chooses her favorite piece. This completes our algorithm. |
3.1 Proof of Correctness of the Main Protocol In this section we analyze the correctness and the complexity of the main protocol. For our proof, we assume that Core and Correction satisfy the properties mentioned earlier. For now, we take as granted the following theorems, which are proved in Sections and respectively. |
Theorem 2 The Core protocol in Section satisfies Core properties and , and makes at most [MATH] cut queries and [MATH] evaluation queries. |
Theorem 3 The Correction protocol in Section satisfies Correction properties and , and makes no queries. We start with one of the most important observations for our analysis: running Core twice with the same cutter creates an edge in the domination graph. Lemma , as well as Lemma below, has its counterpart in |
concerning their core protocol. Lemma 1 Starting with an envy-free partial allocation, if Core is executed two (not necessarily consecutive) times with the same agent, say [MATH] , as the cutter, then in the resulting allocation [MATH] dominates the agent who got the insignificant piece in the first execution. Moreover... |
Proof. If only one piece [MATH] is partially allocated in the first execution, let [MATH] , where [MATH] [MATH] are the allocated and unallocated parts of [MATH] respectively. It holds that [MATH] , and, for all other pieces [MATH] allocated in that execution, [MATH] Therefore, [MATH] is the insignificant piece and is ... |
If more than one piece was partially allocated, then by Core property exactly two pieces were partially allocated. Therefore, the residue after the first execution is [MATH] , where [MATH] [MATH] are the unallocated parts of the insignificant and the other partially allocated piece respectively. By the definition of [M... |
Next, we need the existence of a suitable input for Correction at line of the main protocol. Recall that [MATH] is the allocation output by the [MATH] th execution of Core The following lemma is rather straightforward using a pigeonhole principle argument. |
Lemma 2 Suppose Core is run [MATH] consecutive times with agent [MATH] as the cutter. Then, there exists an allocation [MATH] such that for all agents [MATH] |
[MATH] Proof. It is possible that [MATH] only if [MATH] is the maximum of [MATH] [MATH] Let [MATH] , for [MATH] . It suffices to show that for some [MATH] [MATH] for all [MATH] . This, however, is straightforward since there are at least [MATH] options for [MATH] and [MATH] only takes [MATH] values. |
Finally, the next lemma guarantees that lines do create a second domination over an agent already dominated by agent [MATH] , if such a domination is not already there, without destroying envy-freeness of the overall allocation. |
Lemma 3 Suppose we have an envy free allocation [MATH] where an agent, [MATH] , dominates two other agents, [MATH] and [MATH] . If we run Core with the remaining agent [MATH] as the cutter, excluding [MATH] from competition, then [MATH] will dominate [MATH] or [MATH] in the resulting envy-free allocation. |
Proof. The exclusion of [MATH] from competition greatly simplifies the execution of Core . It is easy to check that in this case Core is equivalent to the following algorithm: |
[MATH] cuts the residue into 4 equal-valued pieces. If agents [MATH] and [MATH] have different favorite pieces, the agents choose a piece in the order [MATH] and the algorithm terminates with a complete envy-free allocation. |
Otherwise, [MATH] and [MATH] make a 2-mark on their common piece. Suppose [MATH] has the rightmost mark on this piece (the other case is symmetric). |
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