On the complexity of choosing the branching literal in DPLL

The DPLL algorithm, developed by Davis and Putnam [DP60] and by Davis, Logemann, and Loveland [DLL62], is the most popular complete algorithm for the problem of satisfiability of a set of propositional clauses (SAT). While it is outperformed by local search algorithms on satisfiable formulas [SLM92, SKC94], it is still used because local search algorithms can give a definite answer only if the formula is satisfiable. If the formula is unsatisfiable, local search algorithm do not prove that the formula is unsatisfiable, but they only return with a “not found” answer. Recent programs using the DPLL algorithm are quite different from the first implementations. First, many optimizations have been discovered, for instance the linear unit propagation algorithm. Second, the heuristics for choosing the next literal to branch on have greatly been improved. Developing better heuristics is crucial, especially for the first steps of the algorithm, as a wrong choice may cause an exponential increase of the running time. There are many papers, in the literature, that focus on the heuristics for choosing the literal to branch in the DPLL algorithm. Hooker and Vinay [HV95] performed a probabilistic and experimental analysis of several heuristics. Usually, when new heuristics are introduced, they are compared with the best ones available at the moment. We are not aware, however, of any work about the general problem of finding the best literal to branch on. Papers in the literature usually analyze particular heuristics, not the general properties of the problem of finding the best literal.