Probabilistic satisfiability: algorithms with the presence and absence of a phase transition
We investigate the role of cycles structures (i.e., subsets of clauses of the form l̄1 ∨ l2, l̄1 ∨ l3, l̄2 ∨ l̄3) in the quality of the lower bound (LB) of modern MaxSAT solvers. Given a cycle structure, we have two options: (i) use the cycle structure just to detect inconsistent subformulas in the underestimation component, and (ii) replace the cycle structure with l̄1, l1 ∨ l̄2 ∨ l̄3, l̄1 ∨ l2 ∨ l3 by applying MaxSAT resolution and, at the same time, change the behaviour of the underestimation component. We first show that it is better to apply MaxSAT resolution to cycle structures occurring in inconsistent subformulas detected using unit propagation or failed literal detection. We then propose a heuristic that guides the application of MaxSAT resolution to cycle structures during failed literal detection, and evaluate this heuristic by implementing it in MaxSatz, obtaining a new solver called MaxSatzc. Our experiments on weighted MaxSAT and Partial MaxSAT instances indicate that MaxSatzc substantially improves MaxSatz on many hard random, crafted and industrial instances.