Stable Matchings with Restricted Preferences: Structure and Complexity

ABSTRACTIt is well known that every stable matching instance I has a rotation poset R(I) that can be computed efficiently and the downsets of R(I) are in one-to-one correspondence with the stable matchings of I. Furthermore, for every poset P, an instance I(P) can be constructed efficiently so that the rotation poset of I(P) is isomorphic to P. In this case, we say that I(P) realizes P. Many researchers exploit the rotation poset of an instance to develop fast algorithms or to establish the hardness of stable matching problems. In order to gain a parameterized understanding of the complexity of sampling stable matchings, Bhatnagar et al.[1] introduced stable matching instances whose preference lists are restricted but nevertheless model situations that arise in practice. In this paper, we study four such parameterized restrictions. Our goal is to characterize the rotation posets that arise from these models: k-bounded, where each agent has at most k acceptable partners; k-attribute, where each agent a has an associated vector ⃗a ∈ Rk, and a ranks agents in decreasing order according to a linear function φa : Rk → R; (k1, k2)-list, where the men and women can be partitioned into k1 and k2 sets (respectively) such that within each set all agents have identical preferences; k-range, where for each woman w, the men's rankings of w differ by at most k-1, and symmetrically for the women's rankings of each man m. We prove that there is a constant k so that every rotation poset is realized by some instance in models 1--3 (k ≥ 3 for k-bounded, k ≥ 6 for k-attribute, and k1 ≥ 2, k2 = ∞ for (k1, k2)-list, respectively). We describe efficient algorithms for constructing such instances given the Hasse diagram of a poset. As a consequence, the fundamental problem of counting stable matchings remains #BIS-complete even for these restricted instances. For k-range preferences, we show that a poset P is realizable if and only if the Hasse diagram of P has pathwidth bounded by functions of k. Using this characterization, we show that the following problems are fixed parameter tractable when parameterized by the range of the instance: exactly counting and uniformly sampling stable matchings, finding median, sex-equal, and balanced stable matchings.