“Almost stable” matchings in the roommates problem

  • Authors:

  • David J. Abraham;Péter Biró;David F. Manlove

  • Affiliations:

  • Computer Science Department, Carnegie-Mellon University;Department of Algebra, and Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Hungary;Department of Computing Science, University of Glasgow, UK

  • Venue:
  • WAOA'05 Proceedings of the Third international conference on Approximation and Online Algorithms
  • Year:
  • 2005

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Abstract

An instance of the classical Stable Roommates problem (sr) need not admit a stable matching. This motivates the problem of finding a matching that is “as stable as possible”, i.e. admits the fewest number of blocking pairs. In this paper we prove that, given an sr instance with n agents, in which all preference lists are complete, the problem of finding a matching with the fewest number of blocking pairs is NP-hard and not approximable within $n^{\frac{1}{2}-\varepsilon}$, for any ε0, unless P=NP. If the preference lists contain ties, we improve this result to n1−ε. Also, we show that, given an integer K and an sr instance I in which all preference lists are complete, the problem of deciding whether I admits a matching with exactly K blocking pairs is NP-complete. By contrast, if K is constant, we give a polynomial-time algorithm that finds a matching with at most (or exactly) K blocking pairs, or reports that no such matching exists. Finally, we give upper and lower bounds for the minimum number of blocking pairs over all matchings in terms of some properties of a stable partition, given an sr instance I.