The Complexity of Finding Fair Many-to-One Matchings

TL;DR

This paper investigates the computational complexity of finding fair many-to-one matchings in bipartite graphs, introducing fixed-parameter algorithms and complexity dichotomies for fairness criteria based on colors.

Contribution

It provides the first fixed-parameter tractable algorithms for fair matchings and establishes complexity boundaries based on colors and degree constraints.

Findings

Finding fair matchings is NP-hard for three colors and degree five.

Developed fixed-parameter algorithms using color coding and ILP techniques.

Proved a new separation theorem related to Hall-like conditions.

Abstract

We analyze the (parameterized) computational complexity of "fair" variants of bipartite many-to-one matching, where each vertex from the "left" side is matched to exactly one vertex and each vertex from the "right" side may be matched to multiple vertices. We want to find a "fair" matching, in which each vertex from the right side is matched to a "fair" set of vertices. Assuming that each vertex from the left side has one color modeling its attribute, we study two fairness criteria. In one of them, we deem a vertex set fair if for any two colors, the difference between the numbers of their occurrences does not exceed a given threshold. Fairness is relevant when finding many-to-one matchings between students and colleges, voters and constituencies, and applicants and firms. Here colors may model sociodemographic attributes, party memberships, and qualifications, respectively. We show that finding a fair many-to-one matching is NP-hard even for three colors and maximum degree five. Our main contribution is the design of fixed-parameter tractable algorithms with respect to the number of vertices on the right side. Our algorithms make use of a variety of techniques including color coding. At the core lie integer linear programs encoding Hall like conditions. To establish the correctness of our integer programs, we prove a new separation result, inspired by Frank's separation theorem [Frank, Discrete Math. 1982], which may also be of independent interest. We further obtain complete complexity dichotomies regarding the number of colors and the maximum degree of each side.