TY - GEN
T1 - Envy-free matchings in bipartite graphs and their applications to fair-division
AU - Aigner-Horev, Elad
AU - Segal Halevi, David Erel
PY - 2021/11/27
Y1 - 2021/11/27
N2 - 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. Weprove 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/3maximin-share allocation of discrete bads among n agents.
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. Weprove 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/3maximin-share allocation of discrete bads among n agents.
U2 - 10.1016/j.ins.2021.11.059
DO - 10.1016/j.ins.2021.11.059
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VL - 587
SP - 164
EP - 187
BT - Workshop of theoretical aspects of fairness, Patras, Greece
ER -