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

Abstract

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 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 N ω 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.

Landscape analysis of constraint satisfaction problems

“…Here, instead, we take the approach of defining an algorithm whose properties can in principle be computed analytically, at least in the same cases for which a static solution is possible. This change of perspective reflects a similar passage from a static to a dynamic approach in the glass literature [18], and has also been pursued in the statistical mechanics of optimization [58]. In this paper we have only stated but not completed the analytical calculation, but we have shown numerically that problems are easy beyond the 'clustering transition', and with the Coloring Problem that in some cases even beyond the 'Kauzmann' one.…”

A Generating Function Method for the Average-Case Analysis of DPLL

“…A precise calculation of the average size of the search space explored by DPLL (and #DPLL, a version of the procedure solving the enumeration problems #SAT and #COL) as a function of the parameters N and α or c is difficult due to the statistical correlations between branches in the search tree resulting from backtracking. Heuristic derivations were nevertheless proposed by Cocco and Monasson based on a 'dynamic annealing' assumption [10,11,13]. Hereafter, using the linearity of expectation, we show that 'dynamic annealing' turns not to be an assumption at all when the expected tree size is concerned.…”

“…To some extent, the present work is an analytical implementation of an idea put forward by Knuth thirty years ago [21,11]. Knuth indeed proposed to estimate the average computational effort required by a backtracking procedure through successive runs of the non-backtracking counterpart, each weighted in an appropriate way [21].…”

“…Knuth indeed proposed to estimate the average computational effort required by a backtracking procedure through successive runs of the non-backtracking counterpart, each weighted in an appropriate way [21]. This weight is, in the language of Section II.B, simply the probability of a branch (given the heuristic under consideration) in #DPLL search tree times 2 S where S is the number of splits [11].…”

Pseudo Expectation: A Tool for Analyzing Local Search Algorithms

“…(A similar approach has been also proposed by S. Cocco and R. Monasson [CM04].) For a given local search algorithm, the approach takes the following two approximation steps.…”