[Paper Review] Envy-free Matchings in Bipartite Graphs and their Applications to Fair Division

Elad Aigner‐Horev, Erel Segal-Halevi|arXiv (Cornell University)|Jan 28, 2019

Game Theory and Voting SystemsEconomics, Econometrics and Finance33 references20 citations

TL;DR

This paper introduces envy-free matchings in bipartite graphs as a novel graph-theoretic concept, proving a unique structural partition (EFM partition) that isolates all envy-free matchings within a subgraph. It presents polynomial-time algorithms to compute maximum-cardinality and minimum-cost envy-free matchings, and applies them to fair division problems such as proportional cake-cutting and maximin-share allocation of discrete goods and bads.

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-floor(2n/3) maximin-share allocation of discrete bads among n agents.

Motivation & Objective

• To formalize and analyze envy-free matchings in bipartite graphs as a standalone graph-theoretic concept.
• To characterize the structural conditions under which non-empty envy-free matchings exist.
• To develop efficient algorithms for computing envy-free matchings of maximum cardinality and minimum cost.
• To apply envy-free matchings to solve fair division problems involving continuous and discrete resources.
• To unify and generalize existing fair division algorithms by revealing hidden envy-free matching structures in prior work.

Proposed method

• Introduce the concept of envy-free matching w.r.t. set X: no unmatched vertex in X is adjacent to any matched vertex in Y.
• Prove a structural theorem (Theorem 1.3) that every bipartite graph admits a unique partition into XL, YL (X-saturated) and XS, YL (Y-path-saturated), such that all envy-free matchings are contained in G[XL, YL].
• Use the EFM partition to reduce the problem of finding envy-free matchings to finding matchings in a subgraph that is X-saturated.
• Apply maximum cardinality matching algorithms (e.g., Hopcroft-Karp) on G[XL, YL] to compute maximum envy-free matchings in O(m√n) time.
• Extend the algorithm to weighted graphs by combining minimum-cost flow techniques with the EFM partition, achieving O(mn + n² log n) time for minimum-cost maximum-cardinality envy-free matchings.
• Apply the framework to fair division: use envy-free matchings to design symmetric proportional cake-cutting, and 1-out-of-(2n−2) and 1-out-of-⌊2n/3⌋ maximin-share allocation algorithms for goods and bads.

Experimental results

Research questions

• RQ1 What structural properties of a bipartite graph guarantee the existence of a non-empty envy-free matching?
• RQ2 Can an envy-free matching of maximum cardinality be computed efficiently in polynomial time?
• RQ3 Can an envy-free matching of maximum cardinality and minimum total cost be computed efficiently in edge-weighted bipartite graphs?
• RQ4 How can envy-free matchings be leveraged to solve fair division problems with continuous and discrete resources?
• RQ5 What is the relationship between envy-free matchings and existing fair division algorithms such as Lone Divider and APX-MMS?

Key findings

• Every bipartite graph admits a unique EFM partition (X = XS ∪ XL, Y = YS ∪ YL) such that all envy-free matchings are contained in G[XL, YL], and G[XS, YS] is Y-path-saturated (containing only the empty matching).
• An envy-free matching of maximum cardinality can be computed in O(m√n) time by finding a maximum matching in the X-saturated subgraph G[XL, YL].
• For edge-weighted graphs, a maximum-cardinality envy-free matching of minimum total weight can be computed in O(mn + n² log n) time.
• The paper provides a symmetric proportional cake-cutting algorithm using envy-free matchings, achieving fairness with minimal assumptions.
• It presents a 1-out-of-(2n−2) maximin-share allocation algorithm for discrete goods and a 1-out-of-⌊2n/3⌋ maximin-share allocation algorithm for discrete bads, both based on envy-free matchings.
• The structural theorem and algorithms generalize and unify prior results, including those from Lone Divider, APX-MMS, and stable matching frameworks, by revealing the underlying envy-free matching structure.