Computing exact solutions of consensus halving and the Borsuk-Ulam theorem

Deligkas, Argyrios; Fearnley, John; Melissourgos, Themistoklis; Spirakis, Paul G.

doi:10.1016/j.jcss.2020.10.006

Cited by 22 publications

(34 citation statements)

References 35 publications

(30 reference statements)

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“…Note that all the steps in the proof can be computed in polynomial time. Thus, as Con-sensusHalving is FIXP-hard [18], we immediately get the following: Corollary 4. PizzaCutting is FIXP-hard.…”

Section: Hardness Resultsmentioning

confidence: 82%

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The complexity of sharing a pizza

Schnider

2021

Preprint

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Assume you have a 2-dimensional pizza with 2n ingredients that you want to share with your friend. For this you are allowed to cut the pizza using several straight cuts, and then give every second piece to your friend. You want to do this fairly, that is, your friend and you should each get exactly half of each ingredient. How many cuts do you need?It was recently shown using topological methods that n cuts always suffice. In this work, we study the computational complexity of finding such n cuts. Our main result is that this problem is PPA-complete when the ingredients are represented as point sets. For this, we give a new proof that for point sets n cuts suffice, which does not use any topological methods.We further prove several hardness results as well as a higher-dimensional variant for the case where the ingredients are well-separated.

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Section: Hardness Resultsmentioning

confidence: 82%

“…Clearly, the construction in the proof above also works for more than n valuation functions. In [18], it was shown that deciding whether n + 1 valuation functions can be bisected with n cuts is ∃R-hard. It thus follows that it is ∃R-hard to decide whether 2n + 2 masses can be bisected by n lines.…”

Section: Hardness Resultsmentioning

confidence: 99%

The complexity of sharing a pizza

Schnider

2021

Preprint

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“…Very recently, Filos-Ratsikas et al [30] strengthened the PPA-hardness result to the case of very simple valuation functions, namely piecewise constant valuations with at most two blocks of value. Deligkas et al [22] studied the computational complexity of the exact version of the problem, and obtained among other results its membership in a newly introduced class BU (for "Borsuk-Ulam" [12]) and its computational hardness for the well-known class FIXP of Etessami and Yannakakis [25]. Very recently, Deligkas et al [21] showed the PPA-completeness of the related Pizza Sharing problem [35], via a reduction from Consensus-Halving.…”

Section: Related Workmentioning

confidence: 99%

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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“…Besides the applications above, the class FIXP also captures the complexity of other problems, such as branching process and context-free grammars [Etessami and Yannakakis, 2010], equilibrium refinements [Etessami et al, 2014;Etessami, 2021], and more recently the complexity of computing a Bayes-Nash equilibrium in the first-price auction with subjective priors [Filos-Ratsikas et al, 2021]. Besides FIXP, there are some other computational classes that capture the complexity of different fixed point problems, namely the classes BU [Deligkas et al, 2021] and BBU [Batziou et al, 2021] which correspond to the Borsuk-Ulam theorem [Borsuk, 1933], and the class HB [Goldberg and Hollender, 2021], which corresponds to the Hairy Ball theorem [Poincaré, 1882].…”

Section: Related Workmentioning

confidence: 99%

FIXP-membership via Convex Optimization: Games, Cakes, and Markets

Filos-Ratsikas¹,

Hansen²,

Høgh³

et al. 2021

Preprint

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We introduce a new technique for proving membership of problems in FIXP -the class capturing the complexity of computing a fixed-point of an algebraic circuit. Our technique constructs a "pseudogate" which can be used as a black box when building FIXP circuits. This pseudogate, which we term the "OPT-gate", can solve most convex optimization problems. Using the OPT-gate, we prove new FIXP-membership results, and we generalize and simplify several known results from the literature on fair division, game theory and competitive markets.In particular, we prove complexity results for two classic problems: computing a market equilibrium in the Arrow-Debreu model with general concave utilities is in FIXP, and computing an envy-free division of a cake with general valuations is FIXP-complete. We further showcase the wide applicability of our technique, by using it to obtain simplified proofs and extensions of known FIXP-membership results for equilibrium computation for various types of strategic games, as well as the pseudomarket mechanism of Hylland and Zeckhauser.

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Computing exact solutions of consensus halving and the Borsuk-Ulam theorem

Cited by 22 publications

References 35 publications

The complexity of sharing a pizza

The complexity of sharing a pizza

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

FIXP-membership via Convex Optimization: Games, Cakes, and Markets

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