Size versus stability in the marriage problem

Abstract

Given an instance I of the classical Stable Marriage problem with Incomplete preference lists (smi), a maximum cardinality matching can be larger than a stable matching. In many large-scale applications of smi, we seek to match as many agents as possible. This motivates the problem of finding a maximum cardinality matching in I that admits the smallest number of blocking pairs (so is "as stable as possible"). We show that this problem is NP-hard and not approximable within n^1−ε, for any ε > 0, unless P=NP, where n is the number of men in I. Further, even if all preference lists are of length at most 3, we show that the problem remains NP-hard and not approximable within δ, for some δ > 1. By contrast, we give a polynomial-time algorithm for the case where the preference lists of one sex are of length at most 2.