Size Versus Stability in the Marriage Problem
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.
KeywordsPerfect Match Vertex Cover Maximum Match Stable Match Preference List
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