Complexity Results on DPLL and Resolution

TL;DR

This paper analyzes the computational complexity of making optimal choices in DPLL and resolution algorithms for propositional satisfiability, establishing hardness results for various decision problems related to their optimality.

Contribution

It extends previous complexity results by proving new hardness bounds for optimal literal selection and proof size determination in DPLL and resolution.

Findings

Optimal literal choice in DPLL is Delta[log]^2-hard.

Optimal literal choice becomes NP^PP-hard with variable subset restrictions.

Determining the size of optimal proofs is CoNP-hard for 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 Delta[log]^2-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 Delta[log]^2-hard if a conjecture we make is true. This problem is CoNP-hard for regular resolution.