The Complexity of Approximately Counting Stable Matchings

TL;DR

This paper explores the computational difficulty of approximately counting stable matchings in various attribute-based models, establishing complexity results and showing that efficient approximation schemes are unlikely in most cases.

Contribution

It proves that the problem remains hard to approximate in the k-attribute setting for k ≥ 2 or 3, but is easy in the 1-attribute dot-product case.

Findings

Counting stable matchings is #P-complete in general.

Approximate counting remains hard for k ≥ 2 or 3.

Counting is easy in the 1-attribute dot-product model.

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 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.