On the approximability of the stable matching problem with ties of size two

TL;DR

This paper analyzes the complexity of the stable matching problem with ties of size two, proving NP-hardness and UGC-hardness for approximation, and provides an improved approximation algorithm with a tight 4/3 factor.

Contribution

It offers a tight analysis of an existing approximation algorithm, improving the approximation factor for the problem with ties of size two.

Findings

Proves NP-hardness of maximum stable matching with ties of size two.

Establishes UGC-hardness for approximation within a factor smaller than 4/3.

Provides an improved 4/3-approximation algorithm with tight analysis.

Abstract

The stable matching problem is one of the central problems of algorithmic game theory. If participants are allowed to have ties, the problem of finding a stable matching of maximum cardinality is an NP-hard problem, even when the ties are of size two. Moreover, in this setting it is UGC-hard to provide an approximation for the maximum cardinality stable matching problem with a constant factor smaller than 4/3. In this paper, we give a tight analysis of an approximation algorithm given by Huang and Kavitha for the maximum cardinality stable matching problem with ties of size two, demonstrating an improved 4/3-approximation factor.