Combinatorica
Dempster's rule of combination is #P-complete (research note)
Artificial Intelligence
The hardest constraint problems: a double phase transition
Artificial Intelligence
Easy problems are sometimes hard
Artificial Intelligence
On the hardness of approximate reasoning
Artificial Intelligence
Artificial Intelligence
Beyond NP: the QSAT phase transition
AAAI '99/IAAI '99 Proceedings of the sixteenth national conference on Artificial intelligence and the eleventh Innovative applications of artificial intelligence conference innovative applications of artificial intelligence
Initial experiments in stochastic satisfiability
AAAI '99/IAAI '99 Proceedings of the sixteenth national conference on Artificial intelligence and the eleventh Innovative applications of artificial intelligence conference innovative applications of artificial intelligence
Stochastic Boolean Satisfiability
Journal of Automated Reasoning
Counting Models Using Connected Components
Proceedings of the Seventeenth National Conference on Artificial Intelligence and Twelfth Conference on Innovative Applications of Artificial Intelligence
Random 3-SAT and BDDs: The Plot Thickens Further
CP '01 Proceedings of the 7th International Conference on Principles and Practice of Constraint Programming
Phase Transitions and Backbones of 3-SAT and Maximum 3-SAT
CP '01 Proceedings of the 7th International Conference on Principles and Practice of Constraint Programming
On some central problems in computational complexity.
On some central problems in computational complexity.
The good old Davis-Putnam procedure helps counting models
Journal of Artificial Intelligence Research
Phase transitions of PP-complete satisfiability problems
IJCAI'01 Proceedings of the 17th international joint conference on Artificial intelligence - Volume 1
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The study of phase transitions in algorithmic problems has revealed that usually the critical value of the constrainedness parameter at which the phase transition occurs coincides with the value at which the average cost of natural solvers for the problem peaks. In particular, this confluence of phase transition and peak cost has been observed for the Boolean satisfiability problem and its variants, where the solver used is a Davis-Putnam-type procedure or a suitable modification of it. Here, we investigate the relationship between phase transitions and peak cost for a family of PP-complete satisfiability problems, where the solver used is a symmetric Threshold Counting Davis-Putnam (TCDP) procedure, i.e., a modification of the Counting Davis-Putnam procedure for computing the number of satisfying assignments of a Boolean formula. Our main experimental finding is that, for each of the PP-complete problems considered, the asymptotic probability of solvability undergoes a phase transition at some critical ratio of clauses to variables, but this critical ratio does not always coincide with the ratio at which the average search cost of the symmetric TCDP procedure peaks. Actually, for some of these problems the peak cost occurs at the boundary or even outside of the interval in which the probability of solvability drops from 0.9 to 0.1, and we analyze why this happens.