Stable Matchings with Restricted Preferences: Structure and Complexity
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by
Christine T. Cheng, Will Rosenbaum
2020
Abstract
It 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.
To make the problem of sampling stable matchings more tractable, Bhatnagar et
al. [SODA 2008] 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,
k-attribute, (k_1, k_2)-list, k-range.
We prove that there is a constant k so that every rotation poset is
realized by some instance in the first three models for some fixed constant
k. 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 parametrized by the range of the instance: exactly counting and
uniformly sampling stable matchings, finding median, sex-equal, and balanced
stable matchings.
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