A Topological Characterization of Modulo-<i>p</i> Arguments and Implications for Necklace Splitting

Filos-Ratsikas, Aris; Hollender, Alexandros; Sotiraki, Katerina; Zampetakis, Manolis

doi:10.1137/1.9781611976465.155

Cited by 15 publications

(23 citation statements)

References 64 publications

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“…For our positive results (i.e., membership in PPA in the white-box model and query complexity upper bounds in the black-box model), we will construct an efficient black-box reduction from -Consensus-Halving to -Tucker (Proposition 3.2). We remark here that a similar reduction already exists in the related literature [31], but only applied to the case of additive valuation functions. The extension to the case of general valuations follows along the same lines, and we provide it here for completeness.…”

Section: Black-box Reductions To and From Consensus-halvingmentioning

confidence: 82%

“…The reduction presented in this section is based on a proof of existence for a generalization of Consensus-Halving with general valuations given in [31]. This existence proof was also used in [31] to provide a reduction for additive valuations. It can easily be extended to work for general valuations as well.…”

Section: Proof Of Proposition 32mentioning

confidence: 99%

“…In other words, I + (− ) = I − ( ) and I − (− ) = I + ( ). For a more formal definition of this encoding, see [31].…”

Section: Proof Of Proposition 32mentioning

confidence: 99%

See 2 more Smart Citations

Two's Company, Three's a Crowd: Consensus-Halving for a Constant Number of Agents

Deligkas

Filos-Ratsikas

Hollender

2021

Proceedings of the 22nd ACM Conference on Economics and Computation

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We consider the -Consensus-Halving problem, in which a set of heterogeneous agents aim at dividing a continuous resource into two (not necessarily contiguous) portions that all of them simultaneously consider to be of approximately the same value (up to ). This problem was recently shown to be PPA-complete, for agents and cuts, even for very simple valuation functions. In a quest to understand the root of the complexity of the problem, we consider the setting where there is only a constant number of agents, and we consider both the computational complexity and the query complexity of the problem.For agents with monotone valuation functions, we show a dichotomy: for two agents the problem is polynomial-time solvable, whereas for three or more agents it becomes PPA-complete. Similarly, we show that for two monotone agents the problem can be solved with polynomially-many queries, whereas for three or more agents, we provide exponential query complexity lower bounds. These results are enabled via an interesting connection to a monotone Borsuk-Ulam problem, which may be of independent interest. For agents with general valuations, we show that the problem is PPA-complete and admits exponential query complexity lower bounds, even for two agents.

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Section: Black-box Reductions To and From Consensus-halvingmentioning

confidence: 82%

Section: Proof Of Proposition 32mentioning

confidence: 99%

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Two's Company, Three's a Crowd: Consensus-Halving for a Constant Number of Agents

Deligkas

Filos-Ratsikas

Hollender

2021

Proceedings of the 22nd ACM Conference on Economics and Computation

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“…The group aspect of fair division has been addressed in a number of recent papers [Manurangsi and Suksompong, 2017;Ghodsi et al, 2018;Suksompong, 2018;Segal-Halevi and Nitzan, 2019;Segal-Halevi and Suksompong, 2019;Kyropoulou et al, 2020]. Most of these papers studied the important fairness notion of envy-freeness: an agent is said to be envy-free if she values the goods allocated to her group at least as much as those allocated to any other group.…”

Section: Introductionmentioning

confidence: 99%

“…When allocating goods among individual agents, an EF1 allocation can be found regardless of the number of agents [Lipton et al, 2004]. However, the picture for group allocation is much less clear, even in the simplest case of two groups.…”

Section: Introductionmentioning

confidence: 99%

Almost Envy-Freeness for Groups: Improved Bounds via Discrepancy Theory

Manurangsi

Suksompong

2021

Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence

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We study the allocation of indivisible goods among groups of agents using well-known fairness notions such as envy-freeness and proportionality. While these notions cannot always be satisfied, we provide several bounds on the optimal relaxations that can be guaranteed. For instance, our bounds imply that when the number of groups is constant and the $n$ agents are divided into groups arbitrarily, there exists an allocation that is envy-free up to $\Theta(\sqrt{n})$ goods, and this bound is tight. Moreover, we show that while such an allocation can be found efficiently, it is NP-hard to compute an allocation that is envy-free up to $o(\sqrt{n})$ goods even when a fully envy-free allocation exists. Our proofs make extensive use of tools from discrepancy theory.

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Envy-free Division of Multi-layered Cakes

Igarashi

Meunier

2022

Web and Internet Economics

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A Topological Characterization of Modulo-p Arguments and Implications for Necklace Splitting

Abstract: A. Hollender is supported by an EPSRC doctoral studentship (Reference 1892947).

Cited by 15 publications

References 64 publications

Two's Company, Three's a Crowd: Consensus-Halving for a Constant Number of Agents

Two's Company, Three's a Crowd: Consensus-Halving for a Constant Number of Agents

Almost Envy-Freeness for Groups: Improved Bounds via Discrepancy Theory

Envy-free Division of Multi-layered Cakes

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