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.
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Biró, P., Manlove, D.F., Mittal, S. (2009). Size Versus Stability in the Marriage Problem. In: Bampis, E., Skutella, M. (eds) Approximation and Online Algorithms. WAOA 2008. Lecture Notes in Computer Science, vol 5426. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-93980-1_2
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DOI: https://doi.org/10.1007/978-3-540-93980-1_2
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