The Rainbow at the End of the Line — A PPAD Formulation of the Colorful Carathéodory Theorem with Applications

Meunier, Frédéric; Mulzer, Wolfgang; Sarrabezolles, Pauline; Stein, Yannik

doi:10.1137/1.9781611974782.87

Cited by 20 publications

(15 citation statements)

References 20 publications

(46 reference statements)

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“…Our proof of this result is entirely different from Bárány's proof of the colorful Carathéodory theorem and relies on a topological mapping degree argument, and thus provides a new topological route to prove this theorem, which is less involved than the deduction recently given by Meunier, Mulzer, Sarrabezolles, and Stein [14] to show that algorithmically finding the configuration whose existence is guaranteed by the colorful Carathéodory theorem is in PPAD, that is, informally speaking, it can be found by a (directed) path-following algorithm. Our method involves a limiting argument and thus does not have immediate algorithmic consequences.…”

Section: Introductionmentioning

confidence: 76%

Colorful Coverings of Polytopes and Piercing Numbers of Colorful d-Intervals

Frick

Zerbib

2019

Combinatorica

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We prove a common strengthening of Bárány's colorful Carathéodory theorem and the KKMS theorem. In fact, our main result is a colorful polytopal KKMS theorem, which extends a colorful KKMS theorem due to Shih and Lee [Math. Ann. 296 (1993), no. 1, 35-61] as well as a polytopal KKMS theorem due to Komiya [Econ. Theory 4 (1994), no. 3, 463-466]. The (seemingly unrelated) colorful Carathéodory theorem is a special case as well. We apply our theorem to establish an upper bound on the piercing number of colorful d-interval hypergraphs, extending earlier results of Tardos [Combinatorica 15 (1995), no. 1, 123-134] and Kaiser [Discrete Comput.

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Section: Introductionmentioning

confidence: 76%

Colorful Coverings of Polytopes and Piercing Numbers of Colorful d-Intervals

Frick

Zerbib

2019

Combinatorica

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“…By contrast, our algorithm performs this update by solving linear programs. 5 4 Another problem with the same complexity status is Colorful Carathéodory [MMSS17]. Showing that these problems are complete for a semantic subclass of PPAD ∩ PLS remains an interesting open question.…”

Section: Introductionmentioning

confidence: 99%

“…Another problem with the same complexity status is Colorful Carathéodory[MMSS17]. Showing that these problems are complete for a semantic subclass of PPAD ∩ PLS remains an interesting open question.5 In and of itself, the perturbation lemma does not lead to a finite-time algorithm for finding fair solutions; Alkan[Alk89] provides an instance wherein the perturbations do not converge.…”

mentioning

confidence: 99%

Fully Polynomial-Time Approximation Schemes for Fair Rent Division

Arunachaleswaran

Barman

Rathi

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the problem of fair rent division that entails splitting the rent and allocating the rooms of an apartment among roommates (agents) in a fair manner. In this setup, a distribution of the rent and an accompanying allocation is said to be fair if it is envy free, i.e., under the imposed rents, no agent has a strictly stronger preference for any other agent's room. The cardinal preferences of the agents are expressed via functions which specify the utilities of the agents for the rooms for every possible room rent/price. While envy-free solutions are guaranteed to exist under reasonably general utility functions, efficient algorithms for finding them were known only for quasilinear utilities. This work addresses this notable gap and develops approximation algorithms for fair rent division with minimal assumptions on the utility functions.Specifically, we show that if the agents have continuous, monotone decreasing, and piecewiselinear utilities, then the fair rent-division problem admits a fully polynomial-time approximation scheme (FPTAS). That is, we develop algorithms that find allocations and prices of the rooms such that for each agent a the utility of the room assigned to it is within a factor of (1 + ε) of the utility of the room most preferred by a. Here, ε > 0 is an approximation parameter, and the running time of the algorithms is polynomial in 1/ε and the input size. In addition, we show that the methods developed in this work provide efficient, truthful mechanisms for special cases of the rent-division problem. Envy-free solutions correspond to equilibria of a two-sided matching market with monetary transfers; hence, this work also provides efficient algorithms for finding approximate equilibria in such markets. We complement the algorithmic results by proving that the fair rent division problem (under continuous, monotone decreasing, and piecewise-linear utilities) lies in the intersection of the complexity classes PPAD and PLS .

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“…Since a Tverberg partition is guaranteed to exist if the cardinality of P is large enough, finding such a partition is a total search problem. In fact, the problem of computing a colorful Carathéodory traversal lies in the complexity class PPAD ∩ PLS [9,11], but no better upper bound on the difficulty of the problem is known. Since Sarkaria's proof can be interpreted as a polynomial-time reduction from the problem of finding a Tverberg partition to the problem of finding a colorful traversal, the same upper bound applies to finding Tverberg partitions.…”

Section: Introductionmentioning

confidence: 99%

No-dimensional Tverberg Theorems and Algorithms

Choudhary¹,

Mulzer²

2019

Preprint

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Tverberg's theorem is a classic result in discrete geometry. It states that for any integer k ≥ 2 and any finite d-dimensional point set P ⊂ R d of at least (d + 1)(k − 1) + 1 points, we can partition P into k subsets whose convex hulls have a non-empty intersection. The computational problem of finding such a partition lies in the complexity class PPAD ∩ PLS, but no hardness results are known. Tverberg's theorem also has a colorful variant: the points in P have colors, and under certain conditions, P can be partitioned into colorful sets, i.e., sets in which each color appears exactly once, such that the convex hulls of the sets intersect. To date, the complexity of the corresponding computational problem has not been resolved.Recently, Adiprasito, Bárány, and Mustafa [SODA 2019] proved a no-dimensional version of Tverberg's theorem, in which the convex hulls of the sets in the partition may intersect in an approximate fashion. This allows it to relax the requirement on the cardinality of P . In fact, they prove a slightly stronger result that is based on the colorful Tverberg theorem. The argument is constructive, but it does not result in a polynomial-time algorithm.Here, we present an alternative proof for a no-dimensional Tverberg theorem that leads to an efficient algorithm to find the partition. More specifically, we show that there is a deterministic algorithm that finds for any set P ⊂ R d of n points and any k ∈ {2, . . . , n} in O(nd log k ) time a partition of P into k subsets such that there is a ball of radius * Supported in part by ERC StG 757609.

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The Rainbow at the End of the Line — A PPAD Formulation of the Colorful Carathéodory Theorem with Applications

Cited by 20 publications

References 20 publications

Colorful Coverings of Polytopes and Piercing Numbers of Colorful d-Intervals

Colorful Coverings of Polytopes and Piercing Numbers of Colorful d-Intervals

Fully Polynomial-Time Approximation Schemes for Fair Rent Division

No-dimensional Tverberg Theorems and Algorithms

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