Improved approximation algorithms for two variants of the stable marriage problem with ties

Abstract

We consider the problem of computing a large stable matching in a bipartite graph $G = (A\cup B, E)$ where each vertex $u \in A\cup B$ ranks its neighbors in an order of preference, perhaps involving ties. Let the matched partner of u in a matching M be M(u). A matching M is said to be stable if there is no edge (a, b) such that a is unmatched or prefers b to M(a) and similarly, b is unmatched or prefers a to M(b). While a stable matching in G can be easily computed in linear time by the Gale–Shapley algorithm, it is known that computing a maximum size stable matching is APX-hard. In this paper we first consider the case when the preference lists of vertices in A are strict while the preference lists of vertices in B may include ties. This case is also APX-hard and the current best approximation ratio known here is 25/17 $\approx 1.4706$ which relies on solving an LP. We improve this ratio to 22/15 $\approx 1.4667$ by a simple linear time algorithm. Here we first compute a half-integral stable matching in $\{0,0.5,1\}^{|E|}$ and then round it to an integral stable matching M. The ratio $|\mathsf {OPT}|/{|M|}$ is bounded via a payment scheme that charges other components in $\mathsf {OPT}\oplus M$ to cover the costs of length-5 augmenting paths. There will be no length-3 augmenting paths here. We next consider the following special case of two-sided ties, where every tie length is 2. This case is known to be UGC-hard to approximate to within 4/3. We show a 10/7 $\approx 1.4286$ approximation algorithm here that runs in linear time.