The complexity of approximately counting stable matchings

  • Authors:

  • Prasad Chebolu;Leslie Ann Goldberg;Russell Martin

  • Affiliations:

  • Department of Computer Science, University of Liverpool;Department of Computer Science, University of Liverpool;Department of Computer Science, University of Liverpool

  • Venue:
  • APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
  • Year:
  • 2010

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Abstract

We investigate the complexity of approximately counting stable matchings in the k-attribute model, where the preference lists are determined by dot products of "preference vectors" with "attribute vectors", or by Euclidean distances between "preference points" and "attribute points". Irving and Leather [16] proved that counting the number of stable matchings in the general case is #P-complete. Counting the number of stable matchings is reducible to counting the number of downsets in a (related) partial order [16] and is interreducible, in an approximation-preserving sense, to a class of problems that includes counting the number of independent sets in a bipartite graph (#BIS) [7]. It is conjectured that no FPRAS exists for this class of problems. We show this approximation-preserving interreducibilty remains even in the restricted k-attribute setting when k ≥ 3 (dot products) or k ≥ 2 (Euclidean distances). Finally, we show it is easy to count the number of stable matchings in the 1-attribute dot-product setting.