On the approximability of the stable marriage problem with one-sided ties

TL;DR

This paper improves the approximation ratio for the NP-hard stable marriage problem with one-sided ties from previous bounds, providing a tighter analysis of an existing algorithm.

Contribution

It offers a refined, tight analysis of Huang and Telikepalli's approximation algorithm, achieving a 13/9-approximation for the problem.

Findings

Improved approximation ratio of 13/9 for the problem.

Analysis is tight, confirming the bound's optimality.

Enhances understanding of approximation limits for stable marriage with ties.

Abstract

The classical stable marriage problem asks for a matching between a set of men and a set of women with no blocking pairs, which are pairs formed by a man and a woman who would both prefer switching from their current status to be paired up together. When both men and women have strict preferences over the opposite group, all stable matchings have the same cardinality, and the famous Gale-Shapley algorithm can be used to find one. Differently, if we allow ties in the preference lists, finding a stable matching of maximum cardinality is an NP-hard problem, already when the ties are one-sided, that is, they appear only in the preferences of one group. For this reason, many researchers have focused on developing approximation algorithm for this problem. In this paper, we give a refined analysis of an approximation algorithm given by Huang and Telikepalli (IPCO14) for the stable marriage problem with one-sided ties, which shows an improved 13/9 -approximation factor for the problem. Interestingly, our analysis is tight.