The Existence of Universally Agreed Fairest Semi-matchings in Any Given Bipartite Graph

Abstract

In a bipartite graph \(G=(U\cup V,E)\) where \(E \subseteq U \times V\), a semi-matching is defined as a set of edges \(M\subseteq E\), such that each vertex in U is incident with exactly one edge in M. Many previous works focus on the problem of 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 many different fairness measures, and they often disagree on the fairness comparison of some semi-matching pairs. In this paper, we prove that for any given bipartite graph, there always exists a (set of equally) fairest semi-matching(s) universally agreed by all the fairness measures. 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 any given bipartite graph. To prove this, we propose a partial order relationship (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 criteria of Majorization [10]. We further show that all widely used fairness measures are in agreement on such a (set of equally) fairest semi-matching(s).