The complexity of approximately counting stable matchings

Stable Matchings with Restricted Preferences: Structure and Complexity

“…Item (1) follows from Theorem 1 because counting stable matchings for is equivalent to counting downsets in R ( ), while the problem of counting downsets in posets is known to be #BIS-complete [12]. 2 In the case of -attribute preferences, Item (1) was also proven by Chebolu et al [5] for any ≥ 3. Our techniques give an alternate-arguably simpler-proof of this result, albeit for the weaker condition ≥ 6.…”

“…While the Gale-Shapley algorithm finds some stable matching efficiently, many computational tasks related to the SMP are known to be computationally hard, such as counting and sampling stable matchings [1,5,19,21] and finding "fair" stable matchings for various notions of fairness [6-8, 13, 18, 20, 22-24, 27, 29, 32]. For these problems, the intractability arises due to the underlying structure of the set of stable matchings.…”

“…Moreover, they show that given any finite poset P, a corresponding instance can be efficiently constructed such that realizes Pi.e. R ( ) is isomorphic to P. Problems such as counting and sampling stable matchings [1,5,21], computing a median stable matching [6,7,32], a balanced stable matching [13] and a sex-equal stable matching [23,29] are then shown to be hard because they correspond to problems on the associated rotation poset that are known to be hard.…”

“…Consequently, a natural Markov Chain Monte Carlo approach to sampling stable matchings takes exponential time to converge. Later, Chebolu et al [5] proved a general hardness result for counting stable matchings in two restricted preference models.…”

“…Bhatnagar et al [1] and Chebolu et al [5] established their hardness results by constructing stable matching instances with restricted preference lists that realized certain families of rotation posets. Our goal is to go a step farther: to characterize the family of rotation posets realized by restricted families of stable matching instances.…”