Achieving Rental Harmony with a Secretive Roommate

Frick, Florian; Houston-Edwards, Kelsey; Meunier, Frédéric

doi:10.1080/00029890.2019.1528127

Cited by 13 publications

(17 citation statements)

References 10 publications

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“…We do not state the version we could get in its full generality and we consider only the most interesting cases p = k − 1 and q = k − 1. The first case is actually that of the secretive roommates version considered in the paper by Frick et al [14]. The second case is the following Survivor rental harmony theorem, which is a new result in this area.…”

Section: Other Applicationsmentioning

confidence: 94%

Multilabeled Versions of Sperner's and Fan's Lemmas and Applications

Meunier¹,

Su²

2019

SIAM J. Appl. Algebra Geometry

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We propose a general technique related to the polytopal Sperner lemma for proving old and new multilabeled versions of Sperner's lemma. A notable application of this technique yields a cake-cutting theorem where the number of players and the number of pieces can be independently chosen. We also prove multilabeled versions of Fan's lemma, a combinatorial analogue of the Borsuk-Ulam theorem, and exhibit applications to fair division and graph coloring.2010 Mathematics Subject Classification. Primary 55M20; Secondary 54H25, 05E45, 91B32.

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Section: Other Applicationsmentioning

confidence: 94%

Multilabeled Versions of Sperner's and Fan's Lemmas and Applications

Meunier¹,

Su²

2019

SIAM J. Appl. Algebra Geometry

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“…al. [6] show that an envy-free rent division can be achieved among n people, even if the preferences of only n − 1 housemates are known.…”

Section: Introductionmentioning

confidence: 99%

Fair division with multiple pieces

Nyman

Zerbib

2020

Discrete Applied Mathematics

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Given a set of p players we consider problems concerning envy-free allocation of collections of k pieces from a given set of goods or chores. We show that if p ≤ n and each player can choose k pieces out of n pieces of a cake, then there exist a division of the cake and an allocation of the pieces where at least p 2(k 2 −k+1) players get their desired k pieces each. We further show that if p ≤ k(n − 1) + 1 and each player can choose k pieces, one from each of k cakes that are divided into n pieces each, then there exist a division of the cakes and allocation of the pieces where at least p 2k(k−1) players get their desired k pieces. Finally we prove that if p ≥ k(n− 1)+ 1 and each player can choose one shift in each of k days that are partitioned into n shifts each, then, given that the salaries of the players are fixed, there exist n(1 + ln k) players covering all the shifts, and moreover, if k = 2 then n players suffice. Our proofs combine topological methods and theorems of Füredi, Lovász and Gallai from hypergraph theory.

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“…Our proof is constructive in a purely logical sense but does not directly admit an algorithm for computing a desired division. Usually, finding such an algorithm in envy-free division problems is a byproduct of applying a Sperner-like theorem, which often times provides a path-following method for approximating the required division (see e.g., Su [17, Section 5] and Frick et al [7,Section 5]). Thus, it would be interesting to find an algorithmic version of our proof, especially using a path-following method.…”

Section: Introductionmentioning

confidence: 99%

Envy-free cake division without assuming the players prefer nonempty pieces

Meunier

Zerbib

2019

Isr. J. Math.

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Consider n players having preferences over the connected pieces of a cake, identified with the interval [0, 1]. A classical theorem, found independently by Stromquist and by Woodall in 1980, ensures that, under mild conditions, it is possible to divide the cake into n connected pieces and assign these pieces to the players in an envy-free manner, i.e, such that no player strictly prefers a piece that has not been assigned to her. One of these conditions, considered as crucial, is that no player is happy with an empty piece. We prove that, even if this condition is not satisfied, it is still possible to get such a division when n is a prime number or is equal to 4. When n is at most 3, this has been previously proved by Erel Segal-Halevi, who conjectured that the result holds for any n. The main step in our proof is a new combinatorial lemma in topology, close to a conjecture by Segal-Halevi and which is reminiscent of the celebrated Sperner lemma: instead of restricting the labels that can appear on each face of the simplex, the lemma considers labelings that enjoy a certain symmetry on the boundary.

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Achieving Rental Harmony with a Secretive Roommate

Cited by 13 publications

References 10 publications

Multilabeled Versions of Sperner's and Fan's Lemmas and Applications

Multilabeled Versions of Sperner's and Fan's Lemmas and Applications

Fair division with multiple pieces

Envy-free cake division without assuming the players prefer nonempty pieces

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