An Algorithm for Stable Matching with Approximation up to the Integrality Gap

Abstract

In the stable matching problem we are given a bipartite graph G = (A ∪ B, E) where A and B represent disjoint groups of agents, each of whom has ordinal preferences over the members of the opposite group. The goal is to find an assignment of agents in one group to those in the other such that no pair of agents prefer each other to their assignees. In this thesis we study the stable matching problem with ties and incomplete preferences. If agents are allowed to have ties and incomplete preferences, computing a stable matching of maximum cardinality is known to be NP-hard. Furthermore, it is known to be NP-hard to achieve a performance guarantee of 33/29 − ε (≈ 1.1379) and UGC-hard to attain that of 4/3 − ε (≈ 1.3333). We present a polynomial-time approximation algorithm with a performance guarantee of (3L − 2)/(2L − 1) where L is the maximum tie length. Our result matches the known lower bound on the integrality gap for the associated LP formulation.