The Complexity of Approximately Counting Stable Matchings

Details The Complexity of Approximately Counting Stable Matchings The Complexity of Approximately Counting Stable Matchings

2010-04-11

arxiv.org/abs/1004.1836v4

TCS 2012

Computer Science

Computational Complexity

Fixed typos, small revisions for clarification, etc

Scientific paper

10.1016/j.tcs.2012.02.029

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 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 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 ($#BIS$). 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 $k \geq 3$ (dot products) or $k \geq 2$ (Euclidean distances). Finally, we show it is easy to count the number of stable matchings in the 1-attribute dot-product setting.

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