## Perfect sampling using bounding chains.(English)Zbl 1052.60057

The basic framework of bounding chains and their application to several interesting problems drawn from statistical mechanics, graph theory and the approximation of NP-complete problems is considered. Successful application of Monte Carlo Markov chain techniques requires to find the mixing time of these chains which is, in general, extremely difficult. Bounding chains give a theoretical and experimental bound on the mixing time. Moreover, the perfect sampling algorithms generate variates exactly from the target distribution without the need to know the mixing time at all.
A technique is applied to a finite Markov chain, denoted by $$M$$, with a state space $$\Omega \subseteq C^V$$ where $$V$$ is the set of dimensions and $$C$$ is the set of colors. The colorings $$c(v) \in C$$ for all $$v \in V$$, which satisfy some preassigned restrictions, are considered. The goal is to generate random variates from the stationary distribution $$\pi$$ on the set of colorings. The bounding chain $$M'$$ with state space $$(2^C)^V$$, where $$2^C$$ is the set of subsets of $$C$$, is defined by the requirement that there exists a coupling $$(X_t,Y_t)$$, $$t=0,1,\dots$$, between $$M$$ and $$M'$$ such that $X_t(v) \in Y_t(v) \quad \forall v \in V \Rightarrow X_{t+1}(v) \in Y_{t+1}(v) \quad \forall v \in V, \quad t=0,1,\dots.$ Here $$X_t$$ is a stochastic process evolving according to $$M$$, so that each $$v \in V$$ is given a single $$c \in C$$ in $$X_t$$, and $$Y_t$$ is a stochastic process evolving according to $$M'$$, so that each $$v \in V$$ is given a subset from $$C$$ in $$Y_t$$. If $$Y_0$$ bounds every state in $$\Omega$$, then when $$Y_t$$ bounds just one state $$x$$ it can be accepted as the variate from the stationary distribution $$\pi$$. The bounding chains are presented for transposition chain on permutations, the hard core gas model, proper colorings of a graph, the antiferromagnetic Potts model and sink free orientations of a graph. Estimations of running time are given.

### MSC:

 60J22 Computational methods in Markov chains 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60J27 Continuous-time Markov processes on discrete state spaces 65C05 Monte Carlo methods 65C40 Numerical analysis or methods applied to Markov chains 82B80 Numerical methods in equilibrium statistical mechanics (MSC2010)
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