Graph-Based Algorithms for Boolean Function Manipulation
IEEE Transactions on Computers
Many hard examples for resolution
Journal of the ACM (JACM)
Information Sciences: an International Journal
The hardest constraint problems: a double phase transition
Artificial Intelligence
Analysis of two simple heuristics on a random instance of k-SAT
Journal of Algorithms
Generating hard satisfiability problems
Artificial Intelligence - Special volume on frontiers in problem solving: phase transitions and complexity
Experimental results on the crossover point in random 3-SAT
Artificial Intelligence - Special volume on frontiers in problem solving: phase transitions and complexity
On the complexity of unsatisfiability proofs for random k-CNF formulas
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
FPGA '99 Proceedings of the 1999 ACM/SIGDA seventh international symposium on Field programmable gate arrays
GRASP: A Search Algorithm for Propositional Satisfiability
IEEE Transactions on Computers
Symbolic model checking using SAT procedures instead of BDDs
Proceedings of the 36th annual ACM/IEEE Design Automation Conference
On the satisfiability and maximum satisfiability of random 3-CNF formulas
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
Setting 2 variables at a time yields a new lower bound for random 3-SAT (extended abstract)
STOC '00 Proceedings of the thirty-second 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
A machine program for theorem-proving
Communications of the ACM
Ordered Binary Decision Diagrams and the Davis-Putnam Procedure
CCL '94 Proceedings of the First International Conference on Constraints in Computational Logics
VIS: A System for Verification and Synthesis
CAV '96 Proceedings of the 8th International Conference on Computer Aided Verification
Resolution and binary decision diagrams cannot simulate each other polynomially
Resolution and binary decision diagrams cannot simulate each other polynomially
EVIDENCE FOR A SATISFIABILITY THRESHOLD FOR RANDOM 3CNF FORMULAS
EVIDENCE FOR A SATISFIABILITY THRESHOLD FOR RANDOM 3CNF FORMULAS
Where the really hard problems are
IJCAI'91 Proceedings of the 12th international joint conference on Artificial intelligence - Volume 1
Random 3-SAT and BDDs: The Plot Thickens Further
CP '01 Proceedings of the 7th International Conference on Principles and Practice of Constraint Programming
In Search of a Phase Transition in the AC-Matching Problem
CP '01 Proceedings of the 7th International Conference on Principles and Practice of Constraint Programming
Computer science can use more science
Communications of the ACM
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This paper presents an experimental investigation of the following questions: how does the average-case complexity of random 3- SAT, understood as a function of the order (number of variables) for fixed density (ratio of number of clauses to order) instances, depend on the density? Is there a phase transition in which the complexity shifts from polynomial to exponential? Is the transition dependent or independent of the solver? To study these questions, we gather median and mean running times for a large collection of random 3-SAT problems while systematically varying both densities and the order of the instances. We use three different complete SAT solvers, embodying very different underlying algorithms: GRASP, CPLEX, and CUDD. We observe new phase transitions for all three solvers, where the median running time shifts from polynomial in the order to exponential. The location of the phase transition appears to be solver-dependent. While GRASP and CUDD shift from polynomial to exponential complexity at a density of about 3.8, CUDD exhibits this transition between densities of 0.1 and 0.5. We believe these experimental observations are important for understanding the computational complexity of random 3-SAT, and can be used as a justification for developing density-aware solvers for 3-SAT.