Improving EFX Guarantees through Rainbow Cycle Number

Chaudhury, Bhaskar Ray; Garg, Jugal; Mehlhorn, Kurt; Mehta, Ruta; Misra, Pranabendu

doi:10.1145/3465456.3467605

Cited by 25 publications

(17 citation statements)

References 3 publications

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“…Extending EFX existence from three to four agents is highly non-trivial. Indeed, Chaudhury et al (2021) discovered an instance with four additive agents in which there exists an EFX allocation with one unallocated item such that no progress can be made based on the lexicographic potential function. We show that one unallocated item is the only possible obstacle to EFX existence in any setting with four agents.…”

Section: Our Resultsmentioning

confidence: 99%

Almost Full EFX Exists for Four Agents

Berger

Cohen

Feldman

et al. 2022

AAAI

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The existence of EFX allocations of goods is a major open problem in fair division, even for additive valuations. The current state of the art is that no setting where EFX allocations are impossible is known, and yet, existence results are known only for very restricted settings, such as: (i) agents with identical valuations, (ii) 2 agents, and (iii) 3 agents with additive valuations. It is also known that EFX exists if one can leave n-1 items unallocated, where n is the number of agents. We develop new techniques that allow us to push the boundaries of the enigmatic EFX problem beyond these known results, and (arguably) to simplify proofs of earlier results. Our main result is that every setting with 4 additive agents admits an EFX allocation that leaves at most a single item unallocated. Beyond our main result, we introduce a new class of valuations, termed nice cancelable, which includes additive, unit-demand, budget-additive and multiplicative valuations, among others. Using our new techniques, we show that both our results and previous results for additive valuations extend to nice cancelable valuations.

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Section: Our Resultsmentioning

confidence: 99%

Almost Full EFX Exists for Four Agents

Berger

Cohen

Feldman

et al. 2022

AAAI

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“…Hence, this literature focuses on special families of graphs, such as path graphs, for which such guarantees can be provided (Bouveret et al 2017;Bilò et al 2019;Bei et al 2021a), and on the computational complexity of the existence of fair connected allocations (Deligkas et al 2021;Greco and Scarcello 2020;Igarashi and Peters 2019). Our goal is to provide approximate fairness guarantees for general graphs, by using the idea of charity, which has been explored recently for fair division without the connectedness constraint (Chaudhury et al 2021b;Caragiannis, Gravin, and Huang 2019;Chaudhury et al 2021a;Berger et al 2021).…”

Section: Related Workmentioning

confidence: 99%

“…While the aforementioned star graph example rules out any reasonably fair partition, note that if we could keep just the hub node unallocated, we could partition the leaf nodes in a highly proportional and balanced manner. In the fair division literature, the idea of keeping a few goods unallocated, termed charity, has been used to achieve fairness guarantees that are even stronger than envy-freeness up to one good without the connectedness constraint (Chaudhury et al 2021b;Caragiannis, Gravin, and Huang 2019;Chaudhury et al 2021a;Berger et al 2021). We borrow this idea and show that charity also helps improve fairness when connectedness is desired.…”

Section: Introductionmentioning

confidence: 99%

A Little Charity Guarantees Fair Connected Graph Partitioning

Caragiannis

Micha

Shah

2022

AAAI

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Motivated by fair division applications, we study a fair connected graph partitioning problem, in which an undirected graph with m nodes must be divided between n agents such that each agent receives a connected subgraph and the partition is fair. We study approximate versions of two fairness criteria: \alpha-proportionality requires that each agent receive a subgraph with at least (1/\alpha)*m/n nodes, and \alpha-balancedness requires that the ratio between the sizes of the largest and smallest subgraphs be at most \alpha. Unfortunately, there exist simple examples in which no partition is reasonably proportional or balanced. To circumvent this, we introduce the idea of charity. We show that by "donating" just n-1 nodes, we can guarantee the existence of 2-proportional and almost 2-balanced partitions (and find them in polynomial time), and that this result is almost tight. More generally, we chart the tradeoff between the size of charity and the approximation of proportionality or balancedness we can guarantee.

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“…They also show that an EFX allocation that leaves at most one item unallocated exists for four agents. Very recently, Chaudhury et al [18] show that a (1 − )-approximate EFX allocation with sublinear number of unallocated goods and high Nash welfare exists. It remains unknown whether similar results hold for the allocation of chores.…”

Section: Other Related Workmentioning

confidence: 99%

Approximately EFX Allocations for Indivisible Chores

Zhou¹,

Wu²

2021

Preprint

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In this paper we study how to fairly allocate a set of m indivisible chores to a group of n agents, each of which has a general additive cost function on the items. Since envy-free (EF) allocation is not guaranteed to exist, we consider the notion of envy-freeness up to any item (EFX). In contrast to the fruitful results regarding the (approximation of) EFX allocations for goods, very little is known for the allocation of chores. Prior to our work, for the allocation of chores, it is known that EFX allocations always exist for two agents, or general number of agents with IDO cost functions. For general instances, no non-trivial approximation result regarding EFX allocation is known. In this paper we make some progress in this direction by showing that for three agents we can always compute a 5-approximation of EFX allocation in polynomial time. For n ≥ 4 agents, our algorithm always computes an allocation that achieves an approximation ratio of O(n 2 ) regarding EFX.

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Improving EFX Guarantees through Rainbow Cycle Number

Cited by 25 publications

References 3 publications

Almost Full EFX Exists for Four Agents

Almost Full EFX Exists for Four Agents

A Little Charity Guarantees Fair Connected Graph Partitioning

Approximately EFX Allocations for Indivisible Chores

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