Complexity results on DPLL and resolution

Abstract

DPLL and resolution are two popular methods for solving the problem of propositional satisfiability. Rather than algorithms, they are families of algorithms, as their behavior depend on some choices they face during execution: DPLL depends on the choice of the literal to branch on; resolution depends on the choice of the pair of clauses to resolve at each step. The complexity of making the optimal choice is analyzed in this paper. Extending previous results, we prove that choosing the optimal literal to branch on in DPLL is ∆ p 2 [log n]-hard, and becomes NP PP -hard if branching is only allowed on a subset of variables. Optimal choice in regular resolution is both NP-hard and coNP-hard. The problem of determining the size of the optimal proofs is also analyzed: it is coNP-hard for DPLL, and ∆ p 2 [log n]-hard if a conjecture we make is true. This problem is coNP-hard for regular resolution.

The effect of structural branching on the efficiency of clause learning SAT solving: An experimental study

“…The complexity of making the optimal branching decision during search in DPLL is studied in [30], with the results that, while the problem for the standard DPLL is not on the first level of the polynomial hierarchy (∆ P 2 [log n]-hard), it may be even harder for restricted-branching DPLL (NP PP -hard, i.e., spanning the whole polynomial hierarchy, under a certain assumption, see [30]). …”