4,725 research outputs found
Matchings and Hamilton Cycles with Constraints on Sets of Edges
The aim of this paper is to extend and generalise some work of Katona on the
existence of perfect matchings or Hamilton cycles in graphs subject to certain
constraints. The most general form of these constraints is that we are given a
family of sets of edges of our graph and are not allowed to use all the edges
of any member of this family. We consider two natural ways of expressing
constraints of this kind using graphs and using set systems.
For the first version we ask for conditions on regular bipartite graphs
and for there to exist a perfect matching in , no two edges of which
form a -cycle with two edges of .
In the second, we ask for conditions under which a Hamilton cycle in the
complete graph (or equivalently a cyclic permutation) exists, with the property
that it has no collection of intervals of prescribed lengths whose union is an
element of a given family of sets. For instance we prove that the smallest
family of -sets with the property that every cyclic permutation of an
-set contains two adjacent pairs of points has size between
and . We also give bounds on the general version of this problem
and on other natural special cases.
We finish by raising numerous open problems and directions for further study.Comment: 21 page
Markov Chains for Sampling Matchings
Markov Chain Monte Carlo algorithms are often used to sample combinatorial structures
such as matchings and independent sets in graphs. A Markov chain is defined
whose state space includes the desired sample space, and which has an appropriate stationary
distribution. By simulating the chain for a sufficiently large number of steps,
we can sample from a distribution arbitrarily close to the stationary distribution. The
number of steps required to do this is known as the mixing time of the Markov chain.
In this thesis, we consider a number of Markov chains for sampling matchings, both
in general and more restricted classes of graphs, and also for sampling independent sets
in claw-free graphs. We apply techniques for showing rapid mixing based on two main
approaches: coupling and conductance. We consider chains using single-site moves,
and also chains using large block moves.
Perfect matchings of bipartite graphs are of particular interest in our community.
We investigate the mixing time of a Markov chain for sampling perfect matchings in
a restricted class of bipartite graphs, and show that its mixing time is exponential in
some instances. For a further restricted class of graphs, however, we can show subexponential
mixing time.
One of the techniques for showing rapid mixing is coupling. The bound on the
mixing time depends on a contraction ratio b. Ideally, b < 1, but in the case b = 1 it
is still possible to obtain a bound on the mixing time, provided there is a sufficiently
large probability of contraction for all pairs of states. We develop a lemma which
obtains better bounds on the mixing time in this case than existing theorems, in the
case where b = 1 and the probability of a change in distance is proportional to the
distance between the two states. We apply this lemma to the Dyer-Greenhill chain for
sampling independent sets, and to a Markov chain for sampling 2D-colourings
A superlinear bound on the number of perfect matchings in cubic bridgeless graphs
Lovasz and Plummer conjectured in the 1970's that cubic bridgeless graphs
have exponentially many perfect matchings. This conjecture has been verified
for bipartite graphs by Voorhoeve in 1979, and for planar graphs by Chudnovsky
and Seymour in 2008, but in general only linear bounds are known. In this
paper, we provide the first superlinear bound in the general case.Comment: 54 pages v2: a short (missing) proof of Lemma 10 was adde
Matchings in Random Biregular Bipartite Graphs
We study the existence of perfect matchings in suitably chosen induced
subgraphs of random biregular bipartite graphs. We prove a result similar to a
classical theorem of Erdos and Renyi about perfect matchings in random
bipartite graphs. We also present an application to commutative graphs, a class
of graphs that are featured in additive number theory.Comment: 30 pages and 3 figures - Latest version has updated introduction and
bibliograph
Computing Unique Maximum Matchings in O(m) time for Konig-Egervary Graphs and Unicyclic Graphs
Let alpha(G) denote the maximum size of an independent set of vertices and
mu(G) be the cardinality of a maximum matching in a graph G. A matching
saturating all the vertices is perfect. If alpha(G) + mu(G) equals the number
of vertices of G, then it is called a Konig-Egervary graph. A graph is
unicyclic if it has a unique cycle.
In 2010, Bartha conjectured that a unique perfect matching, if it exists, can
be found in O(m) time, where m is the number of edges.
In this paper we validate this conjecture for Konig-Egervary graphs and
unicylic graphs. We propose a variation of Karp-Sipser leaf-removal algorithm
(Karp and Spiser, 1981), which ends with an empty graph if and only if the
original graph is a Konig-Egervary graph with a unique perfect matching
obtained as an output as well.
We also show that a unicyclic non-bipartite graph G may have at most one
perfect matching, and this is the case where G is a Konig-Egervary graph.Comment: 10 pages, 5 figure
The Matching Problem in General Graphs is in Quasi-NC
We show that the perfect matching problem in general graphs is in Quasi-NC.
That is, we give a deterministic parallel algorithm which runs in
time on processors. The result is obtained by a
derandomization of the Isolation Lemma for perfect matchings, which was
introduced in the classic paper by Mulmuley, Vazirani and Vazirani [1987] to
obtain a Randomized NC algorithm.
Our proof extends the framework of Fenner, Gurjar and Thierauf [2016], who
proved the analogous result in the special case of bipartite graphs. Compared
to that setting, several new ingredients are needed due to the significantly
more complex structure of perfect matchings in general graphs. In particular,
our proof heavily relies on the laminar structure of the faces of the perfect
matching polytope.Comment: Accepted to FOCS 2017 (58th Annual IEEE Symposium on Foundations of
Computer Science
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