Envy-free matchings in bipartite graphs and their applications to fair division

Elad Aigner-Horev; Erel Segal-Halevi

doi:10.1016/j.ins.2021.11.059

Envy-free matchings in bipartite graphs and their applications to fair division

Elad Aigner-Horev, Erel Segal-Halevi

Ariel University

Research output: Contribution to journal › Article › peer-review

18 Scopus citations

Abstract

A matching in a bipartite graph with parts X and Y is called envy-free, if no unmatched vertex in X is a adjacent to a matched vertex in Y. Every perfect matching is envy-free, but envy-free matchings exist even when perfect matchings do not. We prove that every bipartite graph has a unique partition such that all envy-free matchings are contained in one of the partition sets. Using this structural theorem, we provide a polynomial-time algorithm for finding an envy-free matching of maximum cardinality. For edge-weighted bipartite graphs, we provide a polynomial-time algorithm for finding a maximum-cardinality envy-free matching of minimum total weight. We show how envy-free matchings can be used in various fair division problems with either continuous resources (“cakes”) or discrete ones. In particular, we propose a symmetric algorithm for proportional cake-cutting, an algorithm for 1-out-of-(2n-2) maximin-share allocation of discrete goods, and an algorithm for 1-out-of-⌊2n/3⌋ maximin-share allocation of discrete bads among n agents.

Original language	English
Pages (from-to)	164-187
Number of pages	24
Journal	Information Sciences
Volume	587
DOIs	https://doi.org/10.1016/j.ins.2021.11.059
State	Published - Mar 2022
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2021 Elsevier Inc.

Funding

Erel acknowledges Zur Luria [29], who first provided an existential proof to Corollary 1.1(b), as well as instructive answers by Yuval Filmus, Thomas Klimpel and bof in MathOverflow.com, Max, Vincent Tam and Elmex80s in MathStackExchange.com, and helpful comments by anonymous referees to the WTAF 2019 workshop, the EC 2020 conference, and the Information Sciences journal. This research is partly supported by Israel Science Foundation grant 712/20. Erel acknowledges Zur Luria [29] , who first provided an existential proof to Corollary 1.1(b) , as well as instructive answers by Yuval Filmus, Thomas Klimpel and bof in MathOverflow.com, Max, Vincent Tam and Elmex80s in MathStackExchange.com, and helpful comments by anonymous referees to the WTAF 2019 workshop, the EC 2020 conference, and the Information Sciences journal. This research is partly supported by Israel Science Foundation grant 712/20.

Funders	Funder number
EC 2020 conference
Israel Science Foundation	712/20

Keywords

Bipartite graphs
Cake cutting
Fair division
Maximin share
Maximum matching
Perfect matching

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10.1016/j.ins.2021.11.059

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abstract = "A matching in a bipartite graph with parts X and Y is called envy-free, if no unmatched vertex in X is a adjacent to a matched vertex in Y. Every perfect matching is envy-free, but envy-free matchings exist even when perfect matchings do not. We prove that every bipartite graph has a unique partition such that all envy-free matchings are contained in one of the partition sets. Using this structural theorem, we provide a polynomial-time algorithm for finding an envy-free matching of maximum cardinality. For edge-weighted bipartite graphs, we provide a polynomial-time algorithm for finding a maximum-cardinality envy-free matching of minimum total weight. We show how envy-free matchings can be used in various fair division problems with either continuous resources (“cakes”) or discrete ones. In particular, we propose a symmetric algorithm for proportional cake-cutting, an algorithm for 1-out-of-(2n-2) maximin-share allocation of discrete goods, and an algorithm for 1-out-of-⌊2n/3⌋ maximin-share allocation of discrete bads among n agents.",

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AB - A matching in a bipartite graph with parts X and Y is called envy-free, if no unmatched vertex in X is a adjacent to a matched vertex in Y. Every perfect matching is envy-free, but envy-free matchings exist even when perfect matchings do not. We prove that every bipartite graph has a unique partition such that all envy-free matchings are contained in one of the partition sets. Using this structural theorem, we provide a polynomial-time algorithm for finding an envy-free matching of maximum cardinality. For edge-weighted bipartite graphs, we provide a polynomial-time algorithm for finding a maximum-cardinality envy-free matching of minimum total weight. We show how envy-free matchings can be used in various fair division problems with either continuous resources (“cakes”) or discrete ones. In particular, we propose a symmetric algorithm for proportional cake-cutting, an algorithm for 1-out-of-(2n-2) maximin-share allocation of discrete goods, and an algorithm for 1-out-of-⌊2n/3⌋ maximin-share allocation of discrete bads among n agents.

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Envy-free matchings in bipartite graphs and their applications to fair division

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