We address the problem of sampling colorings of a graph $G$ by Markov chain simulation. For most of the article we restrict attention to proper $q$-colorings of a path on $n$ vertices (in statistical physics terms, the one-dimensional $q$-state Potts model at zero temperature), though in later sections we widen our scope to general ``$H$-colorings'' of arbitrary graphs $G$. Existing theoretical analyses of the mixing time of such simulations relate mainly to a dynamics in which a random vertex is selected for updating at each step. However, experimental work is often carried out using systematic strategies that cycle through coordinates in a deterministic manner, a dynamics sometimes known as systematic scan. The mixing time of systematic scan seems more difficult to analyze than that of random updates, and little is currently known. In this article we go some way toward correcting this imbalance. By adapting a variety of techniques, we derive upper and lower bounds (often tight) on the mixing time of systematic scan. An unusual feature of systematic scan as far as the analysis is concerned is that it fails to be time reversible.
We present an improved coupling technique for analyzing the mixing time of Markov chains. Using our technique, we simplify and extend previous results for sampling colorings and independent sets. Our approach uses properties of the stationary distribution to avoid worst-case configurations which arise in the traditional approach. As an application, we show that for $k/Δ>1.764$, the Glauber dynamics on $k$-colorings of a graph on $n$ vertices with maximum degree $Δ$ converges in $O(n\log n)$ steps, assuming $Δ=Ω(\log n)$ and that the graph is triangle-free. Previously, girth $\ge 5$ was needed.…
Uniform Sampling of Proper Graph Colorings via Soft Coloring and Partial Rejection Sampling
School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）；Institute of Algebra, Dresden University of Technology（德累斯顿工业大学代数研究所）
Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the Erdős-Rényi random graph G(n,d/n). While the average degree in G(n,d/n) is d(1-o(1)), it contains many nodes of degree of order $\log n / \log \log n$. The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n,p) such as uniform coloring or the Ising model, the mixing time of Gibbs sampling is at least $n^{1 + Ω(1 / \log \log n)}$. High degree nodes p…
We present a new notion of probabilistic duality for random variables involving mixture distributions. Using this notion, we show how to implement a highly-parallelizable Gibbs sampler for weakly coupled discrete pairwise graphical models with strictly positive factors that requires almost no preprocessing and is easy to implement. Moreover, we show how our method can be combined with blocking to improve mixing. Even though our method leads to inferior mixing times compared to a sequential Gibbs sampler, we argue that our method is still very useful for large dynamic networks, where factors ar…
We show that the natural Glauber dynamics mixes rapidly and generates a random proper edge-coloring of a graph with maximum degree $Δ$ whenever the number of colors is at least $q\geq (\frac{10}{3} + ε)Δ$, where $ε>0$ is arbitrary and the maximum degree satisfies $Δ\geq C$ for a constant $C = C(ε)$ depending only on $ε$. For edge-colorings, this improves upon prior work \cite{Vig99, CDMPP19} which show rapid mixing when $q\geq (\frac{11}{3}-ε_0 ) Δ$, where $ε_0 \approx 10^{-5}$ is a small fixed constant. At the heart of our proof, we establish a matrix trickle-down theorem, generalizing Oppenh…
For spin systems, such as the $q$-colorings and independent-set models, approximating the partition function in the so-called non-uniqueness region, where the model exhibits long-range correlations, is typically computationally hard for bounded-degree graphs. We present new algorithmic results for approximating the partition function and sampling from the Gibbs distribution for spin systems in the non-uniqueness region on random regular bipartite graphs. We give an $\mathsf{FPRAS}$ for counting $q$-colorings for even $q=O\big(\tfracΔ{\logΔ}\big)$ on almost every $Δ$-regular bipartite graph. Th…
Here we study the problem of sampling random proper colorings of a bounded degree graph. Let $k$ be the number of colors and let $d$ be the maximum degree. In 1999, Vigoda showed that the Glauber dynamics is rapidly mixing for any $k > \frac{11}{6} d$. It turns out that there is a natural barrier at $\frac{11}{6}$, below which there is no one-step coupling that is contractive, even for the flip dynamics. We use linear programming and duality arguments to guide our construction of a better coupling. We fully characterize the obstructions to going beyond $\frac{11}{6}$. These examples turn out t…
This paper is concerned with sampling from the uniform distribution on H-colourings of the n-vertex path using systematic scan Markov chains. An H-colouring of the n-vertex path is a homomorphism from the n-vertex path to some fixed graph H. We show that systematic scan for H-colourings of the n-vertex path mixes in O(log n) scans for any fixed H. This is a significant improvement over the previous bound on the mixing time which was O(n^5) scans. Furthermore we show that for a slightly more restricted family of H (where any two vertices are connected by a 2-edge path) systematic scan also mixe…
We present a randomized algorithm which takes as input an undirected graph $G$ on $n$ vertices with maximum degree $Δ$, and a number of colors $k \geq (8/3 + o_Δ(1))Δ$, and returns -- in expected time $\tilde{O}(nΔ^{2}\log{k})$ -- a proper $k$-coloring of $G$ distributed perfectly uniformly on the set of all proper $k$-colorings of $G$. Notably, our sampler breaks the barrier at $k = 3Δ$ encountered in recent work of Bhandari and Chakraborty [STOC 2020]. We also sketch how to modify our methods to relax the restriction on $k$ to $k \geq (8/3 - ε_0)Δ$ for an absolute constant $ε_0 > 0$. As in t…
A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of $k$-colorings of a graph $G$ on $n$ vertices with maximum degree $Δ$ is rapidly mixing for $k\geΔ+2$. In FOCS 1999, Vigoda showed that the flip dynamics (and therefore also Glauber dynamics) is rapidly mixing for any $k>\frac{11}{6}Δ$. It turns out that there is a natural barrier at $\frac{11}{6}$, below which there is no one-step coupling that is contractive with respect to the Hamming metric, even for the flip dynamics. We use linear programming and duality arguments to fully characteri…
We propose a new type of approximate counting algorithms for the problems of enumerating the number of independent sets and proper colorings in low degree graphs with large girth. Our algorithms are not based on a commonly used Markov chain technique, but rather are inspired by developments in statistical physics in connection with correlation decay properties of Gibbs measures and its implications to uniqueness of Gibbs measures on infinite trees, reconstruction problems and local weak convergence methods. On a negative side, our algorithms provide $ε$-approximations only to the logarithms of…
We study Markov chains for randomly sampling $k$-colorings of a graph with maximum degree $Δ$. Our main result is a polynomial upper bound on the mixing time of the single-site update chain known as the Glauber dynamics for planar graphs when $k=Ω(Δ/\logΔ)$. Our results can be partially extended to the more general case where the maximum eigenvalue of the adjacency matrix of the graph is at most $Δ^{1-\eps}$, for fixed $\eps > 0$. The main challenge when $k \le Δ+ 1$ is the possibility of "frozen" vertices, that is, vertices for which only one color is possible, conditioned on the colors of it…
Bounding chains are a technique that offers three benefits to Markov chain practitioners: a theoretical bound on the mixing time of the chain under restricted conditions, experimental bounds on the mixing time of the chain that are provably accurate and construction of perfect sampling algorithms when used in conjunction with protocols such as coupling from the past. Perfect sampling algorithms generate variates exactly from the target distribution without the need to know the mixing time of a Markov chain at all. We present here the basic theory and use of bounding chains for several chains f…
We study the mixing time of a systematic scan Markov chain for sampling from the uniform distribution on proper 7-colourings of a finite rectangular sub-grid of the infinite square lattice, the grid. A systematic scan Markov chain cycles through finite-size subsets of vertices in a deterministic order and updates the colours assigned to the vertices of each subset. The systematic scan Markov chain that we present cycles through subsets consisting of 2x2 sub-grids and updates the colours assigned to the vertices using a procedure known as heat-bath. We give a computer-assisted proof that this s…
The spectral independence approach of Anari et al. (2020) utilized recent results on high-dimensional expanders of Alev and Lau (2020) and established rapid mixing of the Glauber dynamics for the hard-core model defined on weighted independent sets. We develop the spectral independence approach for colorings, and obtain new algorithmic results for the corresponding counting/sampling problems. Let $α^*\approx 1.763$ denote the solution to $\exp(1/x)=x$ and let $α>α^*$. We prove that, for any triangle-free graph $G=(V,E)$ with maximum degree $Δ$, for all $q\geqαΔ+1$, the mixing time of the Glaub…
For many probability distributions of interest, it is quite difficult to obtain samples efficiently. Often, Markov chains are employed to obtain approximately random samples from these distributions. The primary drawback to traditional Markov chain methods is that the mixing time of the chain is usually unknown, which makes it impossible to determine how close the output samples are to having the target distribution. Here we present a new protocol, the randomness recycler (RR), that overcomes this difficulty. Unlike classical Markov chain approaches, an RR-based algorithm creates samples drawn…
The pairwise influence matrix of Dobrushin has long been used as an analytical tool to bound the rate of convergence of Gibbs sampling. In this work, we use Dobrushin influence as the basis of a practical tool to certify and efficiently improve the quality of a discrete Gibbs sampler. Our Dobrushin-optimized Gibbs samplers (DoGS) offer customized variable selection orders for a given sampling budget and variable subset of interest, explicit bounds on total variation distance to stationarity, and certifiable improvements over the standard systematic and uniform random scan Gibbs samplers. In ou…
We consider the discrete three dimensional scan statistics. Viewed as the maximum of an 1-dependent stationary r.v.'s sequence, we provide approximations and error bounds for the probability distribution of the three dimensional scan statistics. Importance sampling algorithm is used to obtains sharp bounds for the simulation error. Simulation results and comparisons with other approximations are presented for the binomial and Poisson models.
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. We also find efficient counting and sampling algorithms for proper $q$-colorings of random $Δ$-regular bipartite graphs when $q$ is suffici…
