13 research outputs found
The Complexity of Approximately Counting Stable Roommate Assignments
We investigate the complexity of approximately counting stable roommate
assignments in two models: (i) the -attribute model, in which the preference
lists are determined by dot products of "preference vectors" with "attribute
vectors" and (ii) the -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 -attribute model () and the
3-Euclidean model() 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