Socially Fair Matching: Exact and Approximation Algorithms.

Matching problems are some of the most well-studied problems in graph theory and combinatorial optimization, with a variety of theoretical as well as practical motivations. However, in many applications of optimization problems, a “solution” corresponds to real-life decisions that have major impact on humans belonging to diverse groups defined by attributes such as gender, race, or ethnicity. Due to this motivation, the notion of algorithmic fairness has recently emerged to prominence. Depending on specific application, researchers have introduced several notions of fairness. In this paper, we study a problem called Socially Fair Matching, which combines the traditional Minimum Weight Perfect Matching problem with the notion of social fairness that has been studied in clustering literature [Abbasi et al., and Ghadiri et al., FAccT, 2021]. In our problem, the input is an edge-weighted complete bipartite graph, where the bipartition represent two groups of entities. The goal is to find a perfect matching as well as an assignment that assigns the cost of each matched edge to one of its endpoints, such that the maximum of the total cost assigned to either of the two groups is minimized. Unlike Minimum Weight Perfect Matching, we show that Socially Fair Matching is weakly NP-hard. On the positive side, we design a deterministic PTAS for the problem when the edge weights are arbitrary. Furthermore, if the weights are integers and polynomial in the number of vertices, then we give a randomized polynomial-time algorithm that solves the problem exactly. Next, we show that this algorithm can be used to obtain a randomized FPTAS when the weights are arbitrary.