Approximability results for stable marriage problems with ties

  • Authors:

  • Magnús M. Halldórsson;Robert W. Irving;Kazuo Iwama;David F. Manlove;Shuichi Miyazaki;Yasufumi Morita;Sandy Scott

  • Affiliations:

  • Department of Computer Science, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland;Computing Science Department, University of Glasgow, Glasgow G12 8QQ, UK;School of Informatics, Kyoto University, Kyoto 606-8501, Japan;Computing Science Department, University of Glasgow, Glasgow G12 8QQ, UK;Academic Center for Computing and Media Studies, Kyoto University, Kyoto 606-8501, Japan;School of Informatics, Kyoto University, Kyoto 606-8501, Japan;Computing Science Department, University of Glasgow, Glasgow G12 8QQ, UK

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2003

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Abstract

We consider instances of the classical stable marriage problem in which persons may include ties in their preference lists. We show that, in such a setting, strong lower bounds hold for the approximability of each of the problems of finding an egalitarian, minimum regret and sex-equal stable matching. We also consider stable marriage instances in which persons may express unacceptable partners in addition to ties. In this setting, we prove that there are constants δ, δ' such that each of the problems of approximating a maximum and minimum cardinality stable matching within factors of δ, δ' (respectively) is NP-hard, under strong restrictions. We also give an approximation algorithm for both problems that has a performance guarantee expressible in terms of the number of lists with ties. This significantly improves on the best-known previous performance guarantee, for the case that the ties are sparse. Our results have applications to large-scale centralized matching schemes.