Sampling Constraint Satisfaction Solutions in the Local Lemma Regime
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We give a Markov chain based algorithm for sampling almost uniform solutions of constraint satisfaction problems (CSPs). Assuming a canonical setting for the Lovász local lemma, where each constraint is violated by a small number of forbidden local configurations, our sampling algorithm is accurate in a local lemma regime, and the running time is a fixed polynomial whose dependency on n is close to linear, where n is the number of variables. Our main approach is a new technique called states compression, which generalizes the "mark/unmark" paradigm of Moitra (Moitra, JACM, 2019), and can give fast local-lemma-based sampling algorithms. As concrete applications of our technique, we give the current best almost-uniform samplers for hypergraph colorings and for CNF solutions.
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