20 research outputs found
The Reversal Ratio of a Poset
Felsner and Reuter introduced the linear extension diameter of a partially
ordered set , denoted \mbox{led}(\mathbf{P}), as the maximum
distance between two linear extensions of , where distance is
defined to be the number of incomparable pairs appearing in opposite orders
(reversed) in the linear extensions. In this paper, we introduce the reversal
ratio of as the ratio of the linear extension
diameter to the number of (unordered) incomparable pairs. We use probabilistic
techniques to provide a family of posets on at most
elements for which the reversal ratio , where
is a constant. We also examine the questions of bounding the reversal ratio
in terms of order dimension and width.Comment: 10 pages, 2 figures; Accepted for publication in ORDE
Big Ramsey degrees using parameter spaces
We show that the universal homogeneous partial order has finite big Ramsey
degrees and discuss several corollaries. Our proof uses parameter spaces and
the Carlson-Simpson theorem rather than (a strengthening of) the
Halpern-L\"auchli theorem and the Milliken tree theorem, which are the primary
tools used to give bounds on big Ramsey degrees elsewhere (originating from
work of Laver and Milliken).
This new technique has many additional applications. To demonstrate this, we
show that the homogeneous universal triangle-free graph has finite big Ramsey
degrees, thus giving a short proof of a recent result of Dobrinen.Comment: 19 pages, 2 figure
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 page
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
Stable Matchings with Restricted Preferences: Structure and Complexity
In the stable marriage (SM) problem, there are two sets of agentsâtraditionally referred to as men and womenâand each agent has a preference list that ranks (a subset of) agents of the opposite sex. The goal is to find a matching between men and women that is stable in the sense that no man-woman pair mutually prefer each other to their assigned partners. In a seminal work, Gale and Shapley showed that stable matchings always exist, and described an efficient algorithm for finding one.
Irving and Leather defined the rotation poset of an SM instance and showed that it determines the structure of the set of stable matchings of the instance. They further showed that every finite poset can be realized as the rotation poset of some SM instance. Consequently, many problemsâsuch as counting stable matchings and finding certain âfairâ stable matchingsâare computationally intractable (NP-hard) in general.
In this paper, we consider SM instances in which certain restrictions are placed on the preference lists. We show that three natural preference models?k-bounded, k-attribute, and (k1, k2)-listâcan realize arbitrary rotation posets for constant values of k. Hence even in these highly restricted preference models, many stable matching problems remain intractable. In contrast, we show that for any fixed constant k, the rotation posets of k-range instances are highly restricted. As a consequence, we show that exactly counting and uniformly sampling stable matchings, finding median, sex-equal, and balanced stable matchings are fixed-parameter tractable when parameterized by the range of the instance. Thus, these problems can be solved in polynomial time on instances of the k-range model for any fixed constant k