In this paper we consider the problem of computing an "optimal" popular matching. We assume that our input instance G = (A ∪ P, E1 over(∪, ̇) ⋯ over(∪, ̇) Er) admits a popular matching and here we are asked to return not any popular matching but an optimal popular matching, where the definition of optimality is given as a part of the problem statement; for instance, optimality could be fairness in which case we are required to return a fair popular matching. We show an O (n2 + m) algorithm for this problem, assuming that the preference lists are strict, where m is the number of edges in G and n is the number of applicants. © 2009 Elsevier B.V. All rights reserved.
Kavitha, T., & Nasre, M. (2009). Optimal popular matchings. Discrete Applied Mathematics, 157(14), 3181–3186. https://doi.org/10.1016/j.dam.2009.06.004