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Easy Cases of Probabilistic Satisfiability
, 1999
"... The Probabilistic Satisfiability problem (PSAT) can be considered as a probabilistic counterpart of the classical SAT problem. In a PSAT instance, each clause in a CNF formula is assigned a probability of being true; the problem consists in checking the consistency of the assigned probabilities. Act ..."
Abstract

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The Probabilistic Satisfiability problem (PSAT) can be considered as a probabilistic counterpart of the classical SAT problem. In a PSAT instance, each clause in a CNF formula is assigned a probability of being true; the problem consists in checking the consistency of the assigned probabilities. Actually, PSAT turns out to be computationally much harder than SAT, e.g. it remains difficult for some classes of formulas where SAT can be solved in polynomial time. A column generation approach has been proposed in the literature, where the pricing subproblem reduces to a Weighted MaxSAT problem on the original formula. Here we consider some easy cases of PSAT, where it is possible to give a compact representation of the set of consistent probability assignments. We follow two different approaches, based on two different representations of CNF formulas. First we consider a representation based on directed hypergraphs. By extending a wellknown integer programming formulation of the MaxSAT problem, we solve the case in which the hypergraph does not contain cycles. Then we consider the cooccurrence graph associated with a formula. We provide a solution method for the case in which the cooccurrence graph is a partial 2tree.