Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems

Friedland's Lower Matching Conjecture asserts that if $G$ is a $d$--regular bipartite graph on $v(G)=2n$ vertices, and $m_k(G)$ denotes the number of matchings of size $k$, then $$m_k(G)\geq {n \choose k}^2\left(\frac{d-p}{d}\right)^{n(d-p)}(dp)^{np},$$ where $p=\frac{k}{n}$. When $p=1$, this conjecture reduces to a theorem of Schrijver which says that a $d$--regular bipartite graph on $v(G)=2n$ vertices has at least $$\left(\frac{(d-1)^{d-1}}{d^{d-2}}\right)^n$$ perfect matchings. L. Gurvits proved an asymptotic version of the Lower Matching Conjecture, namely he proved that $$\frac{\ln m_k(G)}{v(G)}\geq \frac{1}{2}\left(p\ln \left(\frac{d}{p}\right)+(d-p)\ln \left(1-\frac{p}{d}\right)-2(1-p)\ln (1-p)\right)+o_{v(G)}(1).$$ In this paper, we prove the Lower Matching Conjecture. In fact, we will prove a slightly stronger statement which gives an extra $c_p\sqrt{n}$ factor compared to the conjecture if $p$ is separated away from $0$ and $1$, and is tight up to a constant factor if $p$ is separated away from $1$. We will also give a new proof of Gurvits's and Schrijver's theorems, and we extend these theorems to $(a,b)$--biregular bipartite graphs

Factors of IID on Trees

Classical ergodic theory for integer-group actions uses entropy as a complete invariant for isomorphism of IID (independent, identically distributed) processes (a.k.a. product measures). This theory holds for amenable groups as well. Despite recent spectacular progress of Bowen, the situation for non-amenable groups, including free groups, is still largely mysterious. We present some illustrative results and open questions on free groups, which are particularly interesting in combinatorics, statistical physics, and probability. Our results include bounds on minimum and maximum bisection for random cubic graphs that improve on all past bounds.Comment: 18 pages, 1 figur

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

Bounds for identifying codes in terms of degree parameters

An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If \M(G) denotes the minimum size of an identifying code of a graph $G$, it was conjectured by F. Foucaud, R. Klasing, A. Kosowski and A. Raspaud that there exists a constant $c$ such that if a connected graph $G$ with $n$ vertices and maximum degree $d$ admits an identifying code, then \M(G)\leq n-\tfrac{n}{d}+c. We use probabilistic tools to show that for any $d\geq 3$, \M(G)\leq n-\tfrac{n}{\Theta(d)} holds for a large class of graphs containing, among others, all regular graphs and all graphs of bounded clique number. This settles the conjecture (up to constants) for these classes of graphs. In the general case, we prove \M(G)\leq n-\tfrac{n}{\Theta(d^{3})}. In a second part, we prove that in any graph $G$ of minimum degree $\delta$ and girth at least 5, \M(G)\leq(1+o_\delta(1))\tfrac{3\log\delta}{2\delta}n. Using the former result, we give sharp estimates for the size of the minimum identifying code of random $d$-regular graphs, which is about $\tfrac{\log d}{d}n$

Separation dimension of bounded degree graphs

The 'separation dimension' of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family $\mathcal{F}$ of total orders of the vertices of $G$ such that for any two disjoint edges of $G$, there exists at least one total order in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on $n$ vertices is $\Theta(\log n)$. In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree $d$ is at most $2^{9log^{\star} d} d$. We also demonstrate that the above bound is nearly tight by showing that, for every $d$, almost all $d$-regular graphs have separation dimension at least $\lceil d/2\rceil$.Comment: One result proved in this paper is also present in arXiv:1212.675

Counting hypergraph matchings up to uniqueness threshold

We study the problem of approximately counting matchings in hypergraphs of bounded maximum degree and maximum size of hyperedges. With an activity parameter $\lambda$, each matching $M$ is assigned a weight $\lambda^{|M|}$. The counting problem is formulated as computing a partition function that gives the sum of the weights of all matchings in a hypergraph. This problem unifies two extensively studied statistical physics models in approximate counting: the hardcore model (graph independent sets) and the monomer-dimer model (graph matchings). For this model, the critical activity $\lambda_c= \frac{d^d}{k (d-1)^{d+1}}$ is the threshold for the uniqueness of Gibbs measures on the infinite $(d+1)$-uniform $(k+1)$-regular hypertree. Consider hypergraphs of maximum degree at most $k+1$ and maximum size of hyperedges at most $d+1$. We show that when $\lambda < \lambda_c$, there is an FPTAS for computing the partition function; and when $\lambda = \lambda_c$, there is a PTAS for computing the log-partition function. These algorithms are based on the decay of correlation (strong spatial mixing) property of Gibbs distributions. When $\lambda > 2\lambda_c$, there is no PRAS for the partition function or the log-partition function unless NP$=$RP. Towards obtaining a sharp transition of computational complexity of approximate counting, we study the local convergence from a sequence of finite hypergraphs to the infinite lattice with specified symmetry. We show a surprising connection between the local convergence and the reversibility of a natural random walk. This leads us to a barrier for the hardness result: The non-uniqueness of infinite Gibbs measure is not realizable by any finite gadgets