Proportionally Fair Matching with Multiple Groups

The study of fair algorithms has become mainstream in machine learning and artificial intelligence due to its increasing demand in dealing with biases and discrimination. Along this line, researchers have considered fair versions of traditional optimization problems including clustering, regression, ranking and voting. However, most of the efforts have been channeled into designing heuristic algorithms, which often do not provide any guarantees on the quality of the solution. In this work, 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], where the problem was studied with only two groups. However, in many practical applications, the number of groups -- although 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 groups. We design exact and approximation algorithms achieving reasonable guarantees on the quality of the matching as well as on the time complexity. Our algorithms are also supported by suitable hardness bounds.