Probabilistic analysis of two heuristics for the 3-satisfiability problem
SIAM Journal on Computing
Many hard examples for resolution
Journal of the ACM (JACM)
Information Sciences: an International Journal
Easy problems are sometimes hard
Artificial Intelligence
Analysis of two simple heuristics on a random instance of k-SAT
Journal of Algorithms
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
On the complexity of unsatisfiability proofs for random k-CNF formulas
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Typical random 3-SAT formulae and the satisfiability threshold
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
2+p-SAT: relation of typical-case complexity to the nature of the phase transition
Random Structures & Algorithms - Special issue on statistical physics methods in discrete probability, combinatorics, and theoretical computer science
A machine program for theorem-proving
Communications of the ACM
A sharp threshold in proof complexity
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Rigorous results for random (2 + p)-SAT
Theoretical Computer Science - Phase transitions in combinatorial problems
Results related to threshold phenomena research in satisfiability: lower bounds
Theoretical Computer Science - Phase transitions in combinatorial problems
Lower bounds for random 3-SAT via differential equations
Theoretical Computer Science - Phase transitions in combinatorial problems
Sat2000: Highlights of Satisfiability Research in the Year 2000
Sat2000: Highlights of Satisfiability Research in the Year 2000
Heavy-Tailed Phenomena in Satisfiability and Constraint Satisfaction Problems
Journal of Automated Reasoning
The Probabilistic Analysis of a Greedy Satisfiability Algorithm
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
Experimental results on the crossover point in satisfiability problems
AAAI'93 Proceedings of the eleventh national conference on Artificial intelligence
Empirical hardness models: Methodology and a case study on combinatorial auctions
Journal of the ACM (JACM)
Generating hard satisfiable formulas by hiding solutions deceptiveily
AAAI'05 Proceedings of the 20th national conference on Artificial intelligence - Volume 1
Generating hard satisfiable formulas by hiding solutions deceptively
Journal of Artificial Intelligence Research
A generating function method for the average-case analysis of DPLL
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
Hi-index | 5.23 |
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 (∼ 2Nω 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.