The Complexity of Approximately Counting Stable Matchings

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.