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

Research output: Contribution to journalArticlepeer-review

16 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 languageEnglish
Pages (from-to)164-187
Number of pages24
JournalInformation Sciences
Volume587
DOIs
StatePublished - Mar 2022

Keywords

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

Fingerprint

Dive into the research topics of 'Envy-free matchings in bipartite graphs and their applications to fair division'. Together they form a unique fingerprint.

Cite this