Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2000-11-10
Phys. Rev. E 63, 026702 (2001)
Physics
Condensed Matter
Disordered Systems and Neural Networks
14 pages, 5 figures, to appear in Phys. Rev. E (Feb 2001). v2: minor errors and references corrected
Scientific paper
10.1103/PhysRevE.63.026702
We study a simple and exactly solvable model for the generation of random satisfiability problems. These consist of $\gamma N$ random boolean constraints which are to be satisfied simultaneously by $N$ logical variables. In statistical-mechanics language, the considered model can be seen as a diluted p-spin model at zero temperature. While such problems become extraordinarily hard to solve by local search methods in a large region of the parameter space, still at least one solution may be superimposed by construction. The statistical properties of the model can be studied exactly by the replica method and each single instance can be analyzed in polynomial time by a simple global solution method. The geometrical/topological structures responsible for dynamic and static phase transitions as well as for the onset of computational complexity in local search method are thoroughly analyzed. Numerical analysis on very large samples allows for a precise characterization of the critical scaling behaviour.
Ricci-Tersenghi Federico
Weigt Martin
Zecchina Riccardo
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