Counting Perfect Matchings and the Switch Chain

We examine the problem of exactly or approximately counting all perfect matchings in hereditary classes of nonbipartite graphs. In particular, we consider the switch Markov chain of Diaconis, Graham, and Holmes. We determine the largest hereditary class for which the chain is ergodic, and define a large new hereditary class of graphs for which it is rapidly mixing. We go on to show that the chain has exponential mixing time for a slightly larger class. We also examine the question of ergodicity of the switch chain in an arbitrary graph. Finally, we give exact counting algorithms for three classes

Marathon: An open source software library for the analysis of Markov-Chain Monte Carlo algorithms

In this paper, we consider the Markov-Chain Monte Carlo (MCMC) approach for random sampling of combinatorial objects. The running time of such an algorithm depends on the total mixing time of the underlying Markov chain and is unknown in general. For some Markov chains, upper bounds on this total mixing time exist but are too large to be applicable in practice. We try to answer the question, whether the total mixing time is close to its upper bounds, or if there is a significant gap between them. In doing so, we present the software library marathon which is designed to support the analysis of MCMC based sampling algorithms. The main application of this library is to compute properties of so-called state graphs which represent the structure of Markov chains. We use marathon to investigate the quality of several bounding methods on four well-known Markov chains for sampling perfect matchings and bipartite graph realizations. In a set of experiments, we compute the total mixing time and several of its bounds for a large number of input instances. We find that the upper bound gained by the famous canonical path method is several magnitudes larger than the total mixing time and deteriorates with growing input size. In contrast, the spectral bound is found to be a precise approximation of the total mixing time

Making Markov chains less lazy

The mixing time of an ergodic, reversible Markov chain can be bounded in terms of the eigenvalues of the chain: specifically, the second-largest eigenvalue and the smallest eigenvalue. It has become standard to focus only on the second-largest eigenvalue, by making the Markov chain "lazy". (A lazy chain does nothing at each step with probability at least 1/2, and has only nonnegative eigenvalues.) An alternative approach to bounding the smallest eigenvalue was given by Diaconis and Stroock and Diaconis and Saloff-Coste. We give examples to show that using this approach it can be quite easy to obtain a bound on the smallest eigenvalue of a combinatorial Markov chain which is several orders of magnitude below the best-known bound on the second-largest eigenvalue.Comment: 8 page

On the switch Markov chain for perfect matchings

We study a simple Markov chain, the switch chain, on the set of all perfect matchings in a bipartite graph. This Markov chain was proposed by Diaconis, Graham and Holmes as a possible approach to a sampling problem arising in Statistics. We ask: for which hereditary classes of graphs is the Markov chain ergodic and for which is it rapidly mixing? We provide a precise answer to the ergodicity question and close bounds on the mixing question. We show for the first time that the mixing time of the switch chain is polynomial in the case of monotone graphs, a class that includes examples of interest in the statistical setting

Kasteleyn cokernels

We consider Kasteleyn and Kasteleyn-Percus matrices, which arise in enumerating matchings of planar graphs, up to matrix operations on their rows and columns. If such a matrix is defined over a principal ideal domain, this is equivalent to considering its Smith normal form or its cokernel. Many variations of the enumeration methods result in equivalent matrices. In particular, Gessel-Viennot matrices are equivalent to Kasteleyn-Percus matrices. We apply these ideas to plane partitions and related planar of tilings. We list a number of conjectures, supported by experiments in Maple, about the forms of matrices associated to enumerations of plane partitions and other lozenge tilings of planar regions and their symmetry classes. We focus on the case where the enumerations are round or $q$-round, and we conjecture that cokernels remain round or $q$-round for related ``impossible enumerations'' in which there are no tilings. Our conjectures provide a new view of the topic of enumerating symmetry classes of plane partitions and their generalizations. In particular we conjecture that a $q$-specialization of a Jacobi-Trudi matrix has a Smith normal form. If so it could be an interesting structure associated to the corresponding irreducible representation of \SL(n,\C). Finally we find, with proof, the normal form of the matrix that appears in the enumeration of domino tilings of an Aztec diamond.Comment: 14 pages, 19 in-line figures. Very minor copy correction

The mixing time of the switch Markov chains: a unified approach

Since 1997 a considerable effort has been spent to study the mixing time of switch Markov chains on the realizations of graphic degree sequences of simple graphs. Several results were proved on rapidly mixing Markov chains on unconstrained, bipartite, and directed sequences, using different mechanisms. The aim of this paper is to unify these approaches. We will illustrate the strength of the unified method by showing that on any $P$-stable family of unconstrained/bipartite/directed degree sequences the switch Markov chain is rapidly mixing. This is a common generalization of every known result that shows the rapid mixing nature of the switch Markov chain on a region of degree sequences. Two applications of this general result will be presented. One is an almost uniform sampler for power-law degree sequences with exponent $\gamma>1+\sqrt{3}$. The other one shows that the switch Markov chain on the degree sequence of an Erd\H{o}s-R\'enyi random graph $G(n,p)$ is asymptotically almost surely rapidly mixing if $p$ is bounded away from 0 and 1 by at least $\frac{5\log n}{n-1}$.Comment: Clarification

The Perfect Matching Reconfiguration Problem

We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of length four. We are interested in the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem is PSPACE-complete even for split graphs and for bipartite graphs of bounded bandwidth with maximum degree five. We then investigate polynomial-time solvable cases. Specifically, we prove that the problem is solvable in polynomial time for strongly orderable graphs (that include interval graphs and strongly chordal graphs), for outerplanar graphs, and for cographs (also known as P_4-free graphs). Furthermore, for each yes-instance from these graph classes, we show that a linear number of flip operations is sufficient and we can exhibit a corresponding sequence of flip operations in polynomial time