The Complexity of Approximately Counting Stable Roommate Assignments

TL;DR

This paper explores the computational complexity of approximately counting stable roommate assignments in specific geometric and attribute-based models, establishing their equivalence to well-known hard counting problems and implications for approximation algorithms.

Contribution

The paper proves that counting stable roommate assignments in certain models is as hard as #IS and #SAT, and shows no efficient approximation scheme exists unless NP=RP.

Findings

Counting in these models is interreducible with #IS and #SAT.

No FPRAS exists for counting stable roommate assignments unless NP=RP.

The 1-attribute stable roommate problem has at most two solutions, allowing polynomial-time exact counting.

Abstract

We investigate the complexity of approximately counting stable roommate assignments in two models: (i) the $k$-attribute model, in which the preference lists are determined by dot products of "preference vectors" with "attribute vectors" and (ii) the $k$-Euclidean model, in which the preference lists are determined by the closeness of the "positions" of the people to their "preferred positions". Exactly counting the number of assignments is #P-complete, since Irving and Leather demonstrated #P-completeness for the special case of the stable marriage problem. We show that counting the number of stable roommate assignments in the $k$-attribute model ($k \geq 4$) and the 3-Euclidean model($k \geq 3$) is interreducible, in an approximation-preserving sense, with counting independent sets (of all sizes) (#IS) in a graph, or counting the number of satisfying assignments of a Boolean formula (#SAT). This means that there can be no FPRAS for any of these problems unless NP=RP. As a consequence, we infer that there is no FPRAS for counting stable roommate assignments (#SR) unless NP=RP. Utilizing previous results by the authors, we give an approximation-preserving reduction from counting the number of independent sets in a bipartite graph (#BIS) to counting the number of stable roommate assignments both in the 3-attribute model and in the 2-Euclidean model. #BIS is complete with respect to approximation-preserving reductions in the logically-defined complexity class $#RH\Pi_1$. Hence, our result shows that an FPRAS for counting stable roommate assignments in the 3-attribute model would give an FPRAS for all of $#RH\Pi_1$. We also show that the 1-attribute stable roommate problem always has either one or two stable roommate assignments, so the number of assignments can be determined exactly in polynomial time.