Algorithms for #BIS-hard problems on expander graphs

We give an FPTAS and an efficient sampling algorithm for the high-fugacity hard-core model on bounded-degree bipartite expander graphs and the low-temperature ferromagnetic Potts model on bounded-degree expander graphs. The results apply, for example, to random (bipartite) $\Delta$-regular graphs, for which no efficient algorithms were known for these problems (with the exception of the Ising model) in the non-uniqueness regime of the infinite $\Delta$-regular tree. We also find efficient counting and sampling algorithms for proper $q$-colorings of random $\Delta$-regular bipartite graphs when $q$ is sufficiently small as a function of $\Delta$

Comparing mixing times on sparse random graphs

It is natural to expect that nonbacktracking random walk will mix faster than simple random walks, but so far this has only been proved in regular graphs. To analyze typical irregular graphs, let $G$ be a random graph on $n$ vertices with minimum degree 3 and a degree distribution that has exponential tails. We determine the precise worst-case mixing time for simple random walk on $G$, and show that, with high probability, it exhibits cutoff at time $\mathbf{h}^{-1} \log n$, where $\mathbf{h}$ is the asymptotic entropy for simple random walk on a Galton--Watson tree that approximates $G$ locally. (Previously this was only known for typical starting points.) Furthermore, we show that this asymptotic mixing time is strictly larger than the mixing time of nonbacktracking walk, via a delicate comparison of entropies on the Galton-Watson tree

Algorithms for #BIS-hard problems on expander graphs

We give an FPTAS and an efficient sampling algorithm for the high-fugacity hard-core model on bounded-degree bipartite expander graphs and the low-temperature ferromagnetic Potts model on bounded-degree expander graphs. The results apply, for example, to random (bipartite) Δ-regular graphs, for which no efficient algorithms were known for these problems (with the exception of the Ising model) in the non-uniqueness regime of the infinite Δ-regular tree

Exact thresholds for Ising-Gibbs samplers on general graphs

We establish tight results for rapid mixing of Gibbs samplers for the Ferromagnetic Ising model on general graphs. We show that if $$(d-1)\tanh\beta<1,$$ then there exists a constant C such that the discrete time mixing time of Gibbs samplers for the ferromagnetic Ising model on any graph of n vertices and maximal degree d, where all interactions are bounded by $\beta$, and arbitrary external fields are bounded by $Cn\log n$. Moreover, the spectral gap is uniformly bounded away from 0 for all such graphs, as well as for infinite graphs of maximal degree d. We further show that when $d\tanh\beta<1$, with high probability over the Erdos-Renyi random graph $G(n,d/n)$, it holds that the mixing time of Gibbs samplers is $$n^{1+\Theta({1}/{\log\log n})}.$$ Both results are tight, as it is known that the mixing time for random regular and Erdos-Renyi random graphs is, with high probability, exponential in n when $(d-1)\tanh\beta>1$, and $d\tanh\beta>1$, respectively. To our knowledge our results give the first tight sufficient conditions for rapid mixing of spin systems on general graphs. Moreover, our results are the first rigorous results establishing exact thresholds for dynamics on random graphs in terms of spatial thresholds on trees.Comment: Published in at http://dx.doi.org/10.1214/11-AOP737 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Local Access to Random Walks

For a graph G on n vertices, naively sampling the position of a random walk of at time t requires work ?(t). We desire local access algorithms supporting position_G(t) queries, which return the position of a random walk from some fixed start vertex s at time t, where the joint distribution of returned positions is 1/poly(n) close to those of a uniformly random walk in ?? distance. We first give an algorithm for local access to random walks on a given undirected d-regular graph with O?(1/(1-?)?n) runtime per query, where ? is the second-largest eigenvalue of the random walk matrix of the graph in absolute value. Since random d-regular graphs G(n,d) are expanders with high probability, this gives an O?(?n) algorithm for a graph drawn from G(n,d) whp, which improves on the naive method for small numbers of queries. We then prove that no algorithm with subconstant error given probe access to an input d-regular graph can have runtime better than ?(?n/log(n)) per query in expectation when the input graph is drawn from G(n,d), obtaining a nearly matching lower bound. We further show an ?(n^{1/4}) runtime per query lower bound even with an oblivious adversary (i.e. when the query sequence is fixed in advance). We then show that for families of graphs with additional group theoretic structure, dramatically better results can be achieved. We give local access to walks on small-degree abelian Cayley graphs, including cycles and hypercubes, with runtime polylog(n) per query. This also allows for efficient local access to walks on polylog degree expanders. We show that our techniques apply to graphs with high degree by extending or results to graphs constructed using the tensor product (giving fast local access to walks on degree n^? graphs for any ? ? (0,1]) and Cartesian product

Local resilience and Hamiltonicity Maker-Breaker games in random-regular graphs

For an increasing monotone graph property \mP the \emph{local resilience} of a graph $G$ with respect to \mP is the minimal $r$ for which there exists of a subgraph $H\subseteq G$ with all degrees at most $r$ such that the removal of the edges of $H$ from $G$ creates a graph that does not possesses \mP. This notion, which was implicitly studied for some ad-hoc properties, was recently treated in a more systematic way in a paper by Sudakov and Vu. Most research conducted with respect to this distance notion focused on the Binomial random graph model \GNP and some families of pseudo-random graphs with respect to several graph properties such as containing a perfect matching and being Hamiltonian, to name a few. In this paper we continue to explore the local resilience notion, but turn our attention to random and pseudo-random \emph{regular} graphs of constant degree. We investigate the local resilience of the typical random $d$-regular graph with respect to edge and vertex connectivity, containing a perfect matching, and being Hamiltonian. In particular we prove that for every positive $\epsilon$ and large enough values of $d$ with high probability the local resilience of the random $d$-regular graph, \GND, with respect to being Hamiltonian is at least $(1-\epsilon)d/6$. We also prove that for the Binomial random graph model \GNP, for every positive $\epsilon>0$ and large enough values of $K$, if $p>\frac{K\ln n}{n}$ then with high probability the local resilience of \GNP with respect to being Hamiltonian is at least $(1-\epsilon)np/6$. Finally, we apply similar techniques to Positional Games and prove that if $d$ is large enough then with high probability a typical random $d$-regular graph $G$ is such that in the unbiased Maker-Breaker game played on the edges of $G$, Maker has a winning strategy to create a Hamilton cycle.Comment: 34 pages. 1 figur

The Support of Open Versus Closed Random Walks

A closed random walk of length ? on an undirected and connected graph G = (V,E) is a random walk that returns to the start vertex at step ?, and its properties have been recently related to problems in different mathematical fields, e.g., geometry and combinatorics (Jiang et al., Annals of Mathematics \u2721) and spectral graph theory (McKenzie et al., STOC \u2721). For instance, in the context of analyzing the eigenvalue multiplicity of graph matrices, McKenzie et al. show that, with high probability, the support of a closed random walk of length ? ? 1 is ?(?^{1/5}) on any bounded-degree graph, and leaves as an open problem whether a stronger bound of ?(?^{1/2}) holds for any regular graph. First, we show that the support of a closed random walk of length ? is at least ?(?^{1/2} / ?{log n}) for any regular or bounded-degree graph on n vertices. Secondly, we prove for every ? ? 1 the existence of a family of bounded-degree graphs, together with a start vertex such that the support is bounded by O(?^{1/2}/?{log n}). Besides addressing the open problem of McKenzie et al., these two results also establish a subtle separation between closed random walks and open random walks, for which the support on any regular (or bounded-degree) graph is well-known to be ?(?^{1/2}) for all ? ? 1. For irregular graphs, we prove that even if the start vertex is chosen uniformly, the support of a closed random walk may still be O(log ?). This rules out a general polynomial lower bound in ? for all graphs. Finally, we apply our results on random walks to obtain new bounds on the multiplicity of the second largest eigenvalue of the adjacency matrices of graphs