Fundamentals of Partial Rejection Sampling

Partial Rejection Sampling is an algorithmic approach to obtaining a perfect sample from a specified distribution. The objects to be sampled are assumed to be represented by a number of random variables. In contrast to classical rejection sampling, in which all variables are resampled until a feasible solution is found, partial rejection sampling aims at greater efficiency by resampling only a subset of variables that `go wrong'. Partial rejection sampling is closely related to Moser and Tardos' algorithmic version of the Lov\'asz Local Lemma, but with the additional requirement that a specified output distribution should be met. This article provides a largely self-contained account of the basic form of the algorithm and its analysis

Perfect sampling from spatial mixing

We introduce a new perfect sampling technique that can be applied to general Gibbs distributions and runs in linear time if the correlation decays faster than the neighborhood growth. In particular, in graphs with subexponential neighborhood growth like [Formula: see text] , our algorithm achieves linear running time as long as Gibbs sampling is rapidly mixing. As concrete applications, we obtain the currently best perfect samplers for colorings and for monomer‐dimer models in such graphs

Uniform Sampling through the Lovász Local Lemma

We propose a new algorithmic framework, called `"partial rejection sampling'', to draw samples exactly from a product distribution, conditioned on none of a number of bad events occurring. Our framework builds new connections between the variable framework of the Lovász Local Lemma and some classical sampling algorithms such as the "cycle-popping"' algorithm for rooted spanning trees. Among other applications, we discover new algorithms to sample satisfying assignments of k-CNF formulas with bounded variable occurrences