The existence of universally agreed fairest semi-matchings in any given bipartite graph

Abstract

In a bipartite graph G=(U∪V,E) where E⊆U×V, a semi-matching is defined as a set of edges M⊆E, such that each vertex in U is incident with exactly one edge in M. Many previous works focus on the problem of finding the fairest semi-matchings: ones that assign U-vertices with V-vertices as fairly as possible. In these works, fairness is usually measured according to a specific index. In fact, there exist various fairness measures, and they often disagree on the fairness comparison of some semi-matching pairs. In this paper, we prove that there always exists one (or a set of equally fair) semi-matching(s), universally agreed by all the existing fairness measures, to be the fairest among all the semi-matchings of a given bipartite graph. In other words, given that fairness measures disagree on many comparisons between semi-matchings, they nonetheless are all in agreement on the (set of) fairest semi-matching(s) for a given bipartite graph. To prove this, we propose a partially ordered relationship (named Transfer-based Comparison) among the semi-matchings, showing that the greatest elements always exist in such a partially ordered set. We then show that such greatest elements can guarantee to be the fairest ones under the fairness measure of Majorization [1]. This further indicates that such fairest semi-matchings are agreed by all the fairness measures which are compatible with Majorization. To the best knowledge of us, this is true for all existing fairness measures.