Heuristic average-case analysis of the backtrack resolution of random 3-satisfiability instances

An analysis of the average-case complexity of solving random 3-Satisfiability (SAT) instances with backtrack algorithms is presented. We first interpret previous rigorous works in a unifying framework based on the statistical physics notions of dynamical trajectories, phase diagram and growth process. It is argued that, under the action of the Davis-Putnam-Loveland- Logemann (DPLL) algorithm, 3-SAT instances are turned into 2 + p-SAT instances whose characteristic parameters (ratio α of clauses per variable, fraction p of 3-clauses) can be followed during the operation, and define resolution trajectories. Depending on the location of trajectories in the phase diagram of the 2+p-SAT model, easy (polynomial) or hard (exponential) resolutions are generated. Three regimes are identified, depending on the ratio α of the 3-SAT instance to be solved. Lower satisfiable (sat) phase: for small ratios, DPLL almost surely finds a solution in a time growing linearly with the number N of variables. Upper sat phase: for intermediate ratios, instances are almost surely satisfiable but finding a solution requires exponential time (∼2 ^Nω with ω > 0) with high probability. Unsat phase: for large ratios, there is almost always no solution and proofs of refutation are exponential. An analysis of the growth of the search tree in both upper sat and unsat regimes is presented, and allows us to estimate ω as a function of α. This analysis is based on an exact relationship between the average size of the search tree and the powers of the evolution operator encoding the elementary steps of the search heuristic.