Sampling in Potts Model on Sparse Random Graphs

We study the problem of sampling almost uniform proper q-colorings in sparse Erdos-Renyi random graphs G(n,d/n), a research initiated by Dyer, Flaxman, Frieze and Vigoda [Dyer et al., RANDOM STRUCT ALGOR, 2006]. We obtain a fully polynomial time almost uniform sampler (FPAUS) for the problem provided q>3d+4, improving the current best bound q>5.5d [Efthymiou, SODA, 2014]. Our sampling algorithm works for more generalized models and broader family of sparse graphs. It is an efficient sampler (in the same sense of FPAUS) for anti-ferromagnetic Potts model with activity 03(1-b)d+4. We further identify a family of sparse graphs to which all these results can be extended. This family of graphs is characterized by the notion of contraction function, which is a new measure of the average degree in graphs

Sampling in Uniqueness from the Potts and Random-Cluster Models on Random Regular Graphs

We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the so-called uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers q >= 3 and Delta >= 3, we develop algorithms that produce samples within error o(1) from the q-state Potts model on random Delta-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases. The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes can potentially induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of q and Delta in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value. In the case of the ferromagnetic Potts model, we are able to simplify the algorithm significantly by utilising the random-cluster representation of the model. In particular, we demonstrate that a percolation-type algorithm succeeds in sampling from the random-cluster model with parameters p,q on random Delta-regular graphs for all values of q >= 1 and p<p_c(q,Delta), where p_c(q,Delta) corresponds to a uniqueness threshold for the model on the Delta-regular tree. When restricted to integer values of q, this yields a simplified algorithm for the ferromagnetic Potts model on random Delta-regular graphs

A simple algorithm for sampling colourings of G(N,D/N) up to Gibbs Uniqueness threshold

Approximate random $k$-colouring of a graph G is a well studied problem in computer science and statistical physics. It amounts to constructing a k-colouring of G which is distributed close to Gibbs distribution in polynomial time. Here, we deal with the problem when the underlying graph is an instance of Erdos-Renyi random graph G(n,d/n), where d is a sufficiently large constant. We propose a novel efficient algorithm for approximate random k-colouring G(n,d/n) for any k>(1+\epsilon)d. To be more specific, with probability at least 1-n^{-\Omega(1)} over the input instances G(n,d/n) and for k>(1+\epsilon)d, the algorithm returns a k-colouring which is distributed within total variation distance n^{-\Omega(1)} from the Gibbs distribution of the input graph instance. The algorithm we propose is neither a MCMC one nor inspired by the message passing algorithms proposed by statistical physicists. Roughly the idea is as follows: Initially we remove sufficiently many edges of the input graph. This results in a ``simple graph" which can be $k$-coloured randomly efficiently. The algorithm colours randomly this simple graph. Then it puts back the removed edges one by one. Every time a new edge is put back the algorithm updates the colouring of the graph so that the colouring remains random. The performance of the algorithm depends heavily on certain spatial correlation decay properties of the Gibbs distribution

Counting Solutions to Random CNF Formulas

We give the first efficient algorithm to approximately count the number of solutions in the random $k$-SAT model when the density of the formula scales exponentially with $k$. The best previous counting algorithm was due to Montanari and Shah and was based on the correlation decay method, which works up to densities $(1+o_k(1))\frac{2\log k}{k}$, the Gibbs uniqueness threshold for the model. Instead, our algorithm harnesses a recent technique by Moitra to work for random formulas. The main challenge in our setting is to account for the presence of high-degree variables whose marginal distributions are hard to control and which cause significant correlations within the formula

Sampling Random Colorings of Sparse Random Graphs

We study the mixing properties of the single-site Markov chain known as the Glauber dynamics for sampling $k$-colorings of a sparse random graph $G(n,d/n)$ for constant $d$. The best known rapid mixing results for general graphs are in terms of the maximum degree $\Delta$ of the input graph $G$ and hold when $k>11\Delta/6$ for all $G$. Improved results hold when $k>\alpha\Delta$ for graphs with girth $\geq 5$ and $\Delta$ sufficiently large where $\alpha\approx 1.7632\ldots$ is the root of $\alpha=\exp(1/\alpha)$; further improvements on the constant $\alpha$ hold with stronger girth and maximum degree assumptions. For sparse random graphs the maximum degree is a function of $n$ and the goal is to obtain results in terms of the expected degree $d$. The following rapid mixing results for $G(n,d/n)$ hold with high probability over the choice of the random graph for sufficiently large constant~$d$. Mossel and Sly (2009) proved rapid mixing for constant $k$, and Efthymiou (2014) improved this to $k$ linear in~$d$. The condition was improved to $k>3d$ by Yin and Zhang (2016) using non-MCMC methods. Here we prove rapid mixing when $k>\alpha d$ where $\alpha\approx 1.7632\ldots$ is the same constant as above. Moreover we obtain $O(n^{3})$ mixing time of the Glauber dynamics, while in previous rapid mixing results the exponent was an increasing function in $d$. As in previous results for random graphs our proof analyzes an appropriately defined block dynamics to "hide" high-degree vertices. One new aspect in our improved approach is utilizing so-called local uniformity properties for the analysis of block dynamics. To analyze the "burn-in" phase we prove a concentration inequality for the number of disagreements propagating in large blocks

Approximation via Correlation Decay when Strong Spatial Mixing Fails

Approximate counting via correlation decay is the core algorithmic technique used in the sharp delineation of the computational phase transition that arises in the approximation of the partition function of antiferromagnetic 2-spin models. Previous analyses of correlation-decay algorithms implicitly depended on the occurrence of strong spatial mixing. This, roughly, means that one uses worst-case analysis of the recursive procedure that creates the subinstances. In this paper, we develop a new analysis method that is more refined than the worst-case analysis. We take the shape of instances in the computation tree into consideration and we amortize against certain “bad” instances that are created as the recursion proceeds. This enables us to show correlation decay and to obtain a fully polynomial-time approximation scheme (FPTAS) even when strong spatial mixing fails. We apply our technique to the problem of approximately counting independent sets in hypergraphs with degree upper bound $\Delta$ and with a lower bound $k$ on the arity of hyperedges. Liu and Lin gave an FPTAS for $k\geq2$ and $\Delta\leq5$ (lack of strong spatial mixing was the obstacle preventing this algorithm from being generalized to $\Delta=6$). Our technique gives a tight result for $\Delta=6$, showing that there is an FPTAS for $k\geq3$ and $\Delta\leq6$. The best previously known approximation scheme for $\Delta=6$ is the Markov-chain simulation based fully polynomial-time randomized approximation scheme (FPRAS) of Bordewich, Dyer, and Karpinski, which only works for $k\geq8$. Our technique also applies for larger values of $k$, giving an FPTAS for $k\geq\Delta$. This bound is not substantially stronger than existing randomized results in the literature. Nevertheless, it gives the first deterministic approximation scheme in this regime. Moreover, unlike existing results, it leads to an FPTAS for counting dominating sets in regular graphs with sufficiently large degree. We further demonstrate that in the hypergraph independent set model, approximating the partition function is NP-hard even within the uniqueness regime. Also, approximately counting dominating sets of bounded-degree graphs (without the regularity restriction) is NP-hard