Exponential Lower Bounds for DPLL Algorithms on Satisfiable Random 3-CNF Formulas

Abstract

We consider the performance of a number of DPLL algorithms on random 3-CNF formulas with n variables and m = rn clauses. A long series of papers analyzing so-called “myopic” DPLL algorithms has provided a sequence of lower bounds for their satisfiability threshold. Indeed, for each myopic algorithm ${\mathcal A}$ it is known that there exists an algorithm-specific clause-density, $r_{\mathcal A}$, such that if $r < r_{\mathcal A}$, the algorithm finds a satisfying assignment in linear time. For example, $r_{\mathcal A}$ equals 8/3 = 2.66.. for orderred-dll and 3.003... for generalized unit clause. We prove that for densities well within the provably satisfiable regime, every backtracking extension of either of these algorithms takes exponential time. Specifically, all extensions of orderred-dll take exponential time for r > 2.78 and the same is true for generalized unit clause for all r > 3.1. Our results imply exponential lower bounds for many other myopic algorithms for densities similarly close to the corresponding $r_{\mathcal A}$.