30 research outputs found
On the dimension of posets with cover graphs of treewidth
In 1977, Trotter and Moore proved that a poset has dimension at most
whenever its cover graph is a forest, or equivalently, has treewidth at most
. On the other hand, a well-known construction of Kelly shows that there are
posets of arbitrarily large dimension whose cover graphs have treewidth . In
this paper we focus on the boundary case of treewidth . It was recently
shown that the dimension is bounded if the cover graph is outerplanar (Felsner,
Trotter, and Wiechert) or if it has pathwidth (Bir\'o, Keller, and Young).
This can be interpreted as evidence that the dimension should be bounded more
generally when the cover graph has treewidth . We show that it is indeed the
case: Every such poset has dimension at most .Comment: v4: minor changes made following helpful comments by the referee
Dimension and cut vertices: an application of Ramsey theory
Motivated by quite recent research involving the relationship between the
dimension of a poset and graph-theoretic properties of its cover graph, we show
that for every , if is a poset and the dimension of a subposet
of is at most whenever the cover graph of is a block of the cover
graph of , then the dimension of is at most . We also construct
examples which show that this inequality is best possible. We consider the
proof of the upper bound to be fairly elegant and relatively compact. However,
we know of no simple proof for the lower bound, and our argument requires a
powerful tool known as the Product Ramsey Theorem. As a consequence, our
constructions involve posets of enormous size.Comment: Final published version with updated reference
Tree-width and dimension
Over the last 30 years, researchers have investigated connections between
dimension for posets and planarity for graphs. Here we extend this line of
research to the structural graph theory parameter tree-width by proving that
the dimension of a finite poset is bounded in terms of its height and the
tree-width of its cover graph.Comment: Updates on solutions of problems and on bibliograph
Minors and dimension
It has been known for 30 years that posets with bounded height and with cover
graphs of bounded maximum degree have bounded dimension. Recently, Streib and
Trotter proved that dimension is bounded for posets with bounded height and
planar cover graphs, and Joret et al. proved that dimension is bounded for
posets with bounded height and with cover graphs of bounded tree-width. In this
paper, it is proved that posets of bounded height whose cover graphs exclude a
fixed topological minor have bounded dimension. This generalizes all the
aforementioned results and verifies a conjecture of Joret et al. The proof
relies on the Robertson-Seymour and Grohe-Marx graph structure theorems.Comment: Updated reference
Nowhere Dense Graph Classes and Dimension
Nowhere dense graph classes provide one of the least restrictive notions of
sparsity for graphs. Several equivalent characterizations of nowhere dense
classes have been obtained over the years, using a wide range of combinatorial
objects. In this paper we establish a new characterization of nowhere dense
classes, in terms of poset dimension: A monotone graph class is nowhere dense
if and only if for every and every , posets of height
at most with elements and whose cover graphs are in the class have
dimension .Comment: v4: Minor changes suggested by a refere
Cliquewidth and dimension
We prove that every poset with bounded cliquewidth and with sufficiently
large dimension contains the standard example of dimension as a subposet.
This applies in particular to posets whose cover graphs have bounded treewidth,
as the cliquewidth of a poset is bounded in terms of the treewidth of the cover
graph. For the latter posets, we prove a stronger statement: every such poset
with sufficiently large dimension contains the Kelly example of dimension
as a subposet. Using this result, we obtain a full characterization of the
minor-closed graph classes such that posets with cover graphs in
have bounded dimension: they are exactly the classes excluding
the cover graph of some Kelly example. Finally, we consider a variant of poset
dimension called Boolean dimension, and we prove that posets with bounded
cliquewidth have bounded Boolean dimension.
The proofs rely on Colcombet's deterministic version of Simon's factorization
theorem, which is a fundamental tool in formal language and automata theory,
and which we believe deserves a wider recognition in structural and algorithmic
graph theory
Stable Matchings with Restricted Preferences: Structure and Complexity
In the stable marriage (SM) problem, there are two sets of agents–traditionally referred to as men and women–and 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 problems–such as counting stable matchings and finding certain “fair” stable matchings–are 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)-list–can 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
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