The Complexity of Approximately Counting Stable Roommate Assignments

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 [12]. We show that counting the number of stable roommate assignments in the k-attribute model (#k-attribute SR, k ≥ 4) and the 3-Euclidean model(#k-Euclidean SR, k ≥ 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 [3], 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 3attribute model and in the 2-Euclidean model. #BIS is complete with respect to approximation-preserving reductions in the logically-defined complexity class #RHΠ1. Hence, our result shows that an FPRAS for counting stable roommate assignments in the 3-attribute model would give an Department of Computer Science, University of Liverpool, Ashton Bldg, Ashton St, Liverpool L69 3BX, United Kingdom. Research supported in part by EPSRC Grant EP/F020651/1. Research supported in part by EPSRC Grant EP/I011528/1.