Divisible subdivisions

Alon, Noga; Krivelevich, Michael

doi:10.1002/jgt.22716

Cited by 11 publications

(12 citation statements)

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“…[BBK22] show through a simple argument that this is upper bounded by the permutation rainbow cycle number R p (d), which is defined by introducing an additional constraint in the definition of R(d) that for all i, j, each vertex in V i has exactly one outgoing edge to some vertex in V j (in addition to exactly one incoming edge from some vertex in V j ). In Section 5.2, we show through a simple argument that R p (d) ≤ 2d − 2, thereby also improving the upper bounds of O(d log(d)) in [AK21] and 8d − 1 in [MS21].…”

Section: Introductionmentioning

confidence: 86%

“…with a zero sum (modulo some integer). In particular, [BBK22] show that the rainbow cycle number is a natural generalization of the zero sum problems studied in Alon and Krivelevich [AK21], and Mészáros and Steiner [MS21]. Both papers [AK21,MS21] aim to upper bound the maximum number of vertices of a complete bidirected graph with integer edge labels avoiding a zero sum cycle (modulo d).…”

Section: Introductionmentioning

confidence: 99%

“…In particular, [BBK22] show that the rainbow cycle number is a natural generalization of the zero sum problems studied in Alon and Krivelevich [AK21], and Mészáros and Steiner [MS21]. Both papers [AK21,MS21] aim to upper bound the maximum number of vertices of a complete bidirected graph with integer edge labels avoiding a zero sum cycle (modulo d). [BBK22] show through a simple argument that this is upper bounded by the permutation rainbow cycle number R p (d), which is defined by introducing an additional constraint in the definition of R(d) that for all i, j, each vertex in V i has exactly one outgoing edge to some vertex in V j (in addition to exactly one incoming edge from some vertex in V j ).…”

Section: Introductionmentioning

confidence: 99%

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EFX Allocations: Simplifications and Improvements

Akrami¹,

Chaudhury²,

Garg³

et al. 2022

Preprint

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The existence of EFX allocations is a fundamental open problem in discrete fair division. Given a set of agents and indivisible goods, the goal is to determine the existence of an allocation where no agent envies another following the removal of any single good from the other agent's bundle. Since the general problem has been illusive, progress is made on two fronts: (i) proving existence when the number of agents is small, (ii) proving existence of relaxations of EFX. In this paper, we improve results on both fronts (and simplify in one of the cases).[CGM20] showed the existence of EFX allocations when there are three agents with additive valuation functions. The proof in [CGM20] is long, requires careful and complex case analysis, and does not extend even when one of the agents has a general monotone valuation function. We prove the existence of EFX allocations with three agents, restricting only one agent to have an additive valuation function (the other agents may have any monotone valuation functions). Our proof technique is significantly simpler and shorter than the proof in [CGM20] and therefore more accessible. In particular, it does not use the concepts of champions, champion-graphs, half-bundles (in contrast to the algorithms in [CKMS21, CGM20, CGM + 21]) and envy-graph (in contrast to most algorithms that prove existence of relaxations of envy-freeness, including weaker relaxations like EF1). Our technique also extends to settings when two agents have any monotone valuation function and one agent has an MMS-feasible valuation function (a strict generalization of nice-cancelable valuation functions [BCFF21] which subsumes additive, budget-additive and unit demand valuation functions). This takes us a step closer to resolving the existence of EFX allocations when all three agents have general monotone valuations.Secondly, we consider relaxations of EFX allocations, namely, approximate-EFX allocations and EFX allocations with few unallocated goods (charity). [CGM + 21] showed the existence of (1 − ε)-EFX allocation with O((n/ε) 4 /5 ) charity by establishing a connection to a problem in extremal combinatorics. We improve the result in [CGM + 21] and prove the existence of (1 − ε)-EFX allocations with O((n/ε) 2 /3 ) charity. In fact, our techniques can be used to prove improved upper-bounds on a problem in zero-sum combinatorics introduced by Alon and Krivelevich [AK21, MS21].

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Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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EFX Allocations: Simplifications and Improvements

Akrami¹,

Chaudhury²,

Garg³

et al. 2022

Preprint

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“…Recently, Mészáros and Steiner [MS21] improved the result of Alon and Krivelevich, showing R i (d) ≤ 8d − 1, with further improvements for prime d. In fact, Mészáros and Steiner generalized the result, allowing the labels to come from an arbitrary commutative group of order d. The proof of the main result in [MS21] can be seen as an extension of an incremental construction of [AK21], combined with an intricate inductive argument that makes use of group-theoretic results.…”

mentioning

confidence: 91%

“…More recently, for a given positive integer d, Alon and Krivelevich [AK21] asked for the maximum integer n such that the edges of the complete bidirected graph ↔ K n can be labeled with integers so that there is no zero-sum cycle modulo d. Denoting this quantity by n = R i (d), they showed through an elegant probabilistic argument that R i (d) ∈ O(d log d), with an improvement to R i (d) ∈ O(d) when d is prime. The application considered in [AK21] is finding cycles of length divisible by d in minors of complete graphs. It is easy to see this question as a special case of our fixed-point cycle problem: simply replace every edge-label k by the function x → x + k (mod d).…”

mentioning

confidence: 99%

Fixed-point cycles and EFX allocations

Berendsohn¹,

Boyadzhiyska²,

Kozma³

2022

Preprint

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We study edge-labelings of the complete bidirected graph ↔ K n with functions from the set [d] = {1, . . . , d} to itself. We call a cycle in ↔ K n a fixed-point cycle if composing the labels of its edges results in a map that has a fixed point, and we say that a labeling is fixed-point-free if no fixed-point cycle exists. For a given d, we ask for the largest value of n, denoted R f (d), for which there exists a fixed-point-free labeling of ↔ K n . Determining R f (d) for all d > 0 is a natural Ramsey-type question, generalizing some well-studied zero-sum problems in extremal combinatorics. The problem was recently introduced by Chaudhury, Garg, Mehlhorn, Mehta, and Misra, who proved that d ≤ R f (d) ≤ d 4 + d and showed that the problem has close connections to EFX allocations, a central problem of fair allocation in social choice theory.In this paper we show the improved bound R f (d) ≤ d 2+o(1) , yielding an efficient (1 − ε)-EFX allocation with n agents and O(n 0.67 ) unallocated goods for any constant ε ∈ (0, 1/2]; this improves the bound of O(n 0.8 ) of Chaudhury, Garg, Mehlhorn, Mehta, and Misra.Additionally, we prove the stronger upper bound 2d − 2, in the case where all edge-labels are permulations. A very special case of this problem, that of finding zero-sum cycles in digraphs whose edges are labeled with elements of Z d , was recently considered by Alon and Krivelevich and by Mészáros and Steiner. Our result improves the bounds obtained by these authors and extends them to labelings from an arbitrary (not necessarily commutative) group, while also simplifying the proof.

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Subdivisions with congruence constraints in digraphs of large chromatic number

Steiner

2023

Journal of Graph Theory

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We prove that for every digraph and every assignment of pairs of integers to its arcs there exists an integer such that every digraph with dichromatic number greater than contains a subdivision of in which is subdivided into a directed path of length congruent to modulo , for every . This generalizes to the directed setting the analogous result by Thomassen for undirected graphs, and at the same time yields a novel short proof of his result.

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Divisible subdivisions

Cited by 11 publications

References 11 publications

EFX Allocations: Simplifications and Improvements

EFX Allocations: Simplifications and Improvements

Fixed-point cycles and EFX allocations

Subdivisions with congruence constraints in digraphs of large chromatic number

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