Stable Matchings with Restricted Preferences: Structure and Complexity
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.
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:
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.