scholarly journals Fast Sampling and Counting k -SAT Solutions in the Local Lemma Regime

2021 ◽  
Vol 68 (6) ◽  
pp. 1-42
Author(s):  
Weiming Feng ◽  
Heng Guo ◽  
Yitong Yin ◽  
Chihao Zhang
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Linear Time ◽  
Approximate Counting ◽  
Mcmc Method ◽  
State Spaces ◽  
Local Structures ◽  
Local Lemma ◽  
New Algorithms ◽  
Cnf Formulas

We give new algorithms based on Markov chains to sample and approximately count satisfying assignments to k -uniform CNF formulas where each variable appears at most d times. For any k and d satisfying kd < n o(1) and k ≥ 20 log k + 20 log d + 60, the new sampling algorithm runs in close to linear time, and the counting algorithm runs in close to quadratic time. Our approach is inspired by Moitra (JACM, 2019), which remarkably utilizes the Lovász local lemma in approximate counting. Our main technical contribution is to use the local lemma to bypass the connectivity barrier in traditional Markov chain approaches, which makes the well-developed MCMC method applicable on disconnected state spaces such as SAT solutions. The benefit of our approach is to avoid the enumeration of local structures and obtain fixed polynomial running times, even if k = ω (1) or d = ω (1).

Exponential convergence of adaptive importance sampling for Markov chains

2000 ◽  
Vol 37 (2) ◽  
pp. 342-358 ◽  
Author(s):  
Keith Baggerly ◽  
Dennis Cox ◽  
Rick Picard
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Importance Sampling ◽  
Exponential Convergence ◽  
State Spaces ◽  
Adaptive Importance Sampling ◽  
Finite State ◽  
The Mean ◽  
General Conditions

We consider adaptive importance sampling for a Markov chain with scoring. It is shown that convergence to the zero-variance importance sampling chain for the mean total score occurs exponentially fast under general conditions. These results extend previous work in Kollman (1993) and in Kollman et al. (1999) for finite state spaces.

Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow

1992 ◽  
Vol 1 (4) ◽  
pp. 351-370 ◽  
Author(s):  
Alistair Sinclair
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Statistical Physics ◽  
Flow Problem ◽  
Low Cost ◽  
Multicommodity Flow ◽  
Approximate Counting ◽  
Mixing Rate ◽  
Combinatorial Structures ◽  
Multicommodity Flow Problem

The paper is concerned with tools for the quantitative analysis of finite Markov chains whose states are combinatorial structures. Chains of this kind have algorithmic applications in many areas, including random sampling, approximate counting, statistical physics and combinatorial optimisation. The efficiency of the resulting algorithms depends crucially on the mixing rate of the chain, i.e., the time taken for it to reach its stationary or equilibrium distribution.The paper presents a new upper bound on the mixing rate, based on the solution to a multicommodity flow problem in the Markov chain viewed as a graph. The bound gives sharper estimates for the mixing rate of several important complex Markov chains. As a result, improved bounds are obtained for the runtimes of randomised approximation algorithms for various problems, including computing the permanent of a 0–1 matrix, counting matchings in graphs, and computing the partition function of a ferromagnetic Ising system. Moreover, solutions to the multicommodity flow problem are shown to capture the mixing rate quite closely: thus, under fairly general conditions, a Markov chain is rapidly mixing if and only if it supports a flow of low cost.

scholarly journals Passage-Time Moments for Positively Recurrent Markov Chains

2001 ◽  
Vol 162 ◽  
pp. 169-185
Author(s):  
Tokuzo Shiga ◽  
Akinobu Shimizu ◽  
Takahiro Soshi
Keyword(s):  
Markov Chain ◽  
Convergence Rate ◽  
Markov Chains ◽  
Direct Product ◽  
Transition Probability ◽  
Passage Time ◽  
State Spaces ◽  
Countable State ◽  
Fractional Moments ◽  
Passage Times

Fractional moments of the passage-times are considered for positively recurrent Markov chains with countable state spaces. A criterion of the finiteness of the fractional moments is obtained in terms of the convergence rate of the transition probability to the stationary distribution. As an application it is proved that the passage time of a direct product process of Markov chains has the same order of the fractional moments as that of the single Markov chain.

Exponential convergence of adaptive importance sampling for Markov chains

2000 ◽  
Vol 37 (02) ◽  
pp. 342-358 ◽  
Author(s):  
Keith Baggerly ◽  
Dennis Cox ◽  
Rick Picard
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Importance Sampling ◽  
Exponential Convergence ◽  
State Spaces ◽  
Adaptive Importance Sampling ◽  
Finite State ◽  
The Mean ◽  
General Conditions

We consider adaptive importance sampling for a Markov chain with scoring. It is shown that convergence to the zero-variance importance sampling chain for the mean total score occurs exponentially fast under general conditions. These results extend previous work in Kollman (1993) and in Kollman et al. (1999) for finite state spaces.

scholarly journals Perturbation analysis for denumerable Markov chains with application to queueing models

2004 ◽  
Vol 36 (3) ◽  
pp. 839-853 ◽  
Author(s):  
Eitan Altman ◽  
Konstantin E. Avrachenkov ◽  
Rudesindo Núñez-Queija
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Transition Probabilities ◽  
Perturbation Parameter ◽  
Queueing Models ◽  
Parametric Perturbation ◽  
Steady State Distribution ◽  
State Distribution ◽  
Birth And Death Processes ◽  
State Spaces

We study the parametric perturbation of Markov chains with denumerable state spaces. We consider both regular and singular perturbations. By the latter we mean that transition probabilities of a Markov chain, with several ergodic classes, are perturbed such that (rare) transitions among the different ergodic classes of the unperturbed chain are allowed. Singularly perturbed Markov chains have been studied in the literature under more restrictive assumptions such as strong recurrence ergodicity or Doeblin conditions. We relax these conditions so that our results can be applied to queueing models (where the conditions mentioned above typically fail to hold). Assuming ν-geometric ergodicity, we are able to explicitly express the steady-state distribution of the perturbed Markov chain as a Taylor series in the perturbation parameter. We apply our results to quasi-birth-and-death processes and queueing models.

scholarly journals Perturbation analysis for denumerable Markov chains with application to queueing models

2004 ◽  
Vol 36 (03) ◽  
pp. 839-853 ◽  
Author(s):  
Eitan Altman ◽  
Konstantin E. Avrachenkov ◽  
Rudesindo Núñez-Queija
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Transition Probabilities ◽  
Perturbation Parameter ◽  
Queueing Models ◽  
Parametric Perturbation ◽  
Steady State Distribution ◽  
State Distribution ◽  
Birth And Death Processes ◽  
State Spaces

We study the parametric perturbation of Markov chains with denumerable state spaces. We consider both regular and singular perturbations. By the latter we mean that transition probabilities of a Markov chain, with several ergodic classes, are perturbed such that (rare) transitions among the different ergodic classes of the unperturbed chain are allowed. Singularly perturbed Markov chains have been studied in the literature under more restrictive assumptions such as strong recurrence ergodicity or Doeblin conditions. We relax these conditions so that our results can be applied to queueing models (where the conditions mentioned above typically fail to hold). Assumingν-geometric ergodicity, we are able to explicitly express the steady-state distribution of the perturbed Markov chain as a Taylor series in the perturbation parameter. We apply our results to quasi-birth-and-death processes and queueing models.

scholarly journals Sampling Random Chordal Graphs by MCMC (Student Abstract)

2020 ◽  
Vol 34 (10) ◽  
pp. 13929-13930
Author(s):  
Wenbo Sun ◽  
Ivona Bezáková
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chains ◽  
Chordal Graph ◽  
Chordal Graphs ◽  
Random Generation ◽  
Mcmc Method ◽  
Bounded Treewidth ◽  
Graph Class ◽  
General Graphs

Chordal graphs are a widely studied graph class, with applications in several areas of computer science, including structural learning of Bayesian networks. Many problems that are hard on general graphs become solvable on chordal graphs. The random generation of instances of chordal graphs for testing these algorithms is often required. Nevertheless, there are only few known algorithms that generate random chordal graphs, and, as far as we know, none of them generate chordal graphs uniformly at random (where each chordal graph appears with equal probability). In this paper we propose a Markov chain Monte Carlo (MCMC) method to sample connected chordal graphs uniformly at random. Additionally, we propose a Markov chain that generates connected chordal graphs with a bounded treewidth uniformly at random. Bounding the treewidth parameter (which bounds the largest clique) has direct implications on the running time of various algorithms on chordal graphs. For each of the proposed Markov chains we prove that they are ergodic and therefore converge to the uniform distribution. Finally, as initial evidence that the Markov chains have the potential to mix rapidly, we prove that the chain on graphs with bounded treewidth mixes rapidly for trees (chordal graphs with treewidth bound of one).

Asymptotic behaviour of sample weighted circuits representing recurrent Markov chains

1990 ◽  
Vol 27 (03) ◽  
pp. 545-556 ◽  
Author(s):  
S. Kalpazidou
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
Probabilistic Interpretation ◽  
Stationary Markov Chain

The asymptotic behaviour of the sequence (𝒞 n (ω), wc,n (ω)/n), is studied where 𝒞 n (ω) is the class of all cycles c occurring along the trajectory ωof a recurrent strictly stationary Markov chain (ξ n ) until time n and wc,n (ω) is the number of occurrences of the cycle c until time n. The previous sequence of sample weighted classes converges almost surely to a class of directed weighted cycles (𝒞∞, ω c ) which represents uniquely the chain (ξ n ) as a circuit chain, and ω c is given a probabilistic interpretation.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

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