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 depends 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 article. Extending previous results, we prove that choosing the optimal literal to branch on in DPLL is Δ 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 Δ p 2 [log n ]-hard if a conjecture we make is true. This problem is coNP-hard for regular resolution.