6 research outputs found
A Simply Exponential Upper Bound on the Maximum Number of Stable Matchings
Stable matching is a classical combinatorial problem that has been the
subject of intense theoretical and empirical study since its introduction in
1962 in a seminal paper by Gale and Shapley. In this paper, we provide a new
upper bound on , the maximum number of stable matchings that a stable
matching instance with men and women can have. It has been a
long-standing open problem to understand the asymptotic behavior of as
, first posed by Donald Knuth in the 1970s. Until now the best
lower bound was approximately , and the best upper bound was . In this paper, we show that for all , for some
universal constant . This matches the lower bound up to the base of the
exponent. Our proof is based on a reduction to counting the number of downsets
of a family of posets that we call "mixing". The latter might be of independent
interest
Stable Matchings with Restricted Preferences: Structure and Complexity
It is well known that every stable matching instance has a rotation poset
that can be computed efficiently and the downsets of are in
one-to-one correspondence with the stable matchings of . Furthermore, for
every poset , an instance can be constructed efficiently so that the
rotation poset of is isomorphic to . In this case, we say that
realizes . 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. [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: -bounded, -attribute, -list,
-range.
We prove that there is a constant so that every rotation poset is
realized by some instance in the first three models for some fixed constant
. 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 -range preferences, we show that a poset is realizable if and only
if the Hasse diagram of has pathwidth bounded by functions of . 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.Comment: Various updates and improvements in response to reviewer comment