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).

This work is supported by NSF Grant CNS-1149661.

Appendix

Claim

For any pair of semi-matchings $M_x$ and $M_y$, there always exists a sequence containing of a Simplest Transfer Sequence and a sequence of Cyclic Neutral Transfer, which changes $M_x$ to $M_y$.

Proof

The notion $M_x \oplus M_y$ denotes the symmetric difference of edges set $M_x$ and $M_y$, that is, $M_x \oplus M_y = (M_x \setminus M_y) \cup (M_y\setminus M_x)$. Let S represents the set of all Cyclic Neutral Transfers in $M_x \oplus M_y$. Suppose $M_{x'}$ can be derived by applying all Cyclic Neutral Transfers of S to $M_x$.

We assume all the edges of $M_{x'} \setminus M_y$ are colored by green, and all the edges of $M_y \setminus M_{x'}$ are colored by red. An observation on $M_{x'} \oplus M_y$ is that there exist one or more V-vertices which are endpoints of only green edges, but not red edges. We call those V-vertices as Starting Vertices. We build a Transfer which is an alternating green-red sequence of edges, as follows. (1) Find a green edge of which the V-endpoint is one arbitrary Starting Vertex, and set its U-endpoint as Current Vertex. Then repeat the following two steps. (2) Find a red edge of which the U-endpoint is Current Vertex. Set its V-endpoint as Current Vertex. (3) Find a green edge of which the V-endpoint is Current Vertex. Set its U-endpoint as Current Vertex. Continue until we cannot find any green edges. Delete all chosen edges from $M_{x'} \oplus M_y$, and then repeat above procedures to build more Transfers until $M_{x'} \oplus M_y$ becomes empty.

Throughout this process, we maintain that among all the obtained Transfers, the source vertex of one Transfer cannot be the sink vertex of another Transfer. Then, an arbitrarily ordered sequence of all the obtained Transfers constructs a Simplest Transfer Sequence from $M_{x'}$ to $M_y$. An illustration of the Transfers Construction is shown in Fig. 3.    $\square$

Fig. 3.

An illustration of Transfers Construction.