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

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

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

Matching random colored points with rectangles

Let S ¿ [0, 1]2 be a set of n points, randomly and uniformly selected. Let R ¿ B be a random partition, or coloring, of S in which each point of S is included in R uniformly at random with probability 1/2. We study the random number M(n) of points of S that are covered by the rectangles of a maximum strong matching of S with axis-aligned rectangles. The matching consists of closed rectangles that cover exactly two points of S of the same color. A matching is strong if all its rectangles are pairwise disjoint. We prove that almost surely M(n) = 0.83 n for n large enough. Our approach is based on modeling a deterministic greedy matching algorithm, that runs over the random point set, as a Markov chain.Postprint (published version

Reversibility and further properties of FCFS infinite bipartite matching

The model of FCFS infinite bipartite matching was introduced in caldentey-kaplan-weiss 2009. In this model there is a sequence of items that are chosen i.i.d. from $\mathcal{C}=\{c_1,\ldots,c_I\}$ and an independent sequence of items that are chosen i.i.d. from $\mathcal{S}=\{s_1,\ldots,s_J\}$, and a bipartite compatibility graph $G$ between $\mathcal{C}$ and $\mathcal{S}$. Items of the two sequences are matched according to the compatibility graph, and the matching is FCFS, each item in the one sequence is matched to the earliest compatible unmatched item in the other sequence. In adan-weiss 2011 a Markov chain associated with the matching was analyzed, a condition for stability was verified, a product form stationary distribution was derived and the rates $r_{c_i,s_j}$ of matches between compatible types $c_i$ and $s_j$ were calculated. In the current paper, we present several new results that unveil the fundamental structure of the model. First, we provide a pathwise Loynes' type construction which enables to prove the existence of a unique matching for the model defined over all the integers. Second, we prove that the model is dynamically reversible: we define an exchange transformation in which we interchange the positions of each matched pair, and show that the items in the resulting permuted sequences are again independent and i.i.d., and the matching between them is FCFS in reversed time. Third, we obtain product form stationary distributions of several new Markov chains associated with the model. As a by product, we compute useful performance measures, for instance the link lengths between matched items.Comment: 33 pages, 12 figure