  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.
