Combinatorial Results on Directed Hypergraphs for the SAT Problem
Directed hypergraphs have already been shown to unveil several combinatorial inspired results for the SAT problem. In this paper we approach the SAT problem by searching a transversal of the directed hypergraphs associated to its instance. We introduce some particular clause orderings and study their influence on the backtrack process, exhibiting a new subclass of CNF for which SAT is polynomial. Based on unit resolution and a novel dichotomous search, a new DPLL-like algorithm and a renaming-based combinatorial approach are proposed. We then investigate the study of weak transversals in this setting and reveal a new degree of a CNF formula unsatisfiability and a structural result about unsatisfiable formulae.
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