Characterization of Super-stable Matchings

TL;DR

This paper studies super-stable matchings in bipartite graphs with ties, providing compact representations, a rotation poset, and a polyhedral characterization, thus advancing understanding of their structure and enumeration.

Contribution

It introduces two compact representations for all super-stable matchings, constructs an explicit rotation poset, and proves the integrality of the super-stable matching polytope.

Findings

Two $O(m)$ size representations for super-stable matchings

Construction of a rotation poset in $O(mn)$ time

Proof that the super-stable matching polytope is integral

Abstract

An instance of the super-stable matching problem with incomplete lists and ties is an undirected bipartite graph $G = (A \cup B, E)$, with an adjacency list being a linearly ordered list of ties. Ties are subsets of vertices equally good for a given vertex. An edge $(x,y) \in E \backslash M$ is a blocking edge for a matching $M$ if by getting matched to each other neither of the vertices $x$ and $y$ would become worse off. Thus, there is no disadvantage if the two vertices would like to match up. A matching $M$ is super-stable if there is no blocking edge with respect to $M$. It has previously been shown that super-stable matchings form a distributive lattice and the number of super-stable matchings can be exponential in the number of vertices. We give two compact representations of size $O(m)$ that can be used to construct all super-stable matchings, where $m$ denotes the number of edges in the graph. The construction of the second representation takes $O(mn)$ time, where $n$ denotes the number of vertices in the graph, and gives an explicit rotation poset similar to the rotation poset in the classical stable marriage problem. We also give a polyhedral characterisation of the set of all super-stable matchings and prove that the super-stable matching polytope is integral, thus solving an open problem stated in the book by Gusfield and Irving .