Proportionally Fair Matching with Multiple Groups

We study matching problems with the notion of proportional fairness. Proportional fairness is one of the most popular notions of group fairness where every group is represented up to an extent proportional to the final selection size. Matching with proportional fairness or more commonly, proportionally fair matching, was introduced in [Chierichetti et al., AISTATS, 2019]. In this problem, we are given a graph G whose edges are colored with colors from a set C. The task is for given 0 <= alpha <= beta <= 1, to find a maximum (alpha, beta)-balanced matching M in G, that is a matching where for every color c is an element of C the number of edges in M of color c is between alpha|M| and beta|M|. Chierichetti et al. initiated the study of this problem with two colors and in the context of bipartite graphs only. However, in many practical applications, the number of colorsalthough often a small constant-is larger than two. In this work, we make the first step towards understanding the computational complexity of proportionally fair matching with more than two colors. We design exact and approximation algorithms achieving reasonable guarantees on the quality of the matching as well as on the time complexity, and our algorithms work in general graphs. Our algorithms are also supported by suitable hardness bounds.