1. Witness of unsatisfiability for a random 3-satisfiability formula.
- Author
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Wu, Lu-Lu, Zhou, Hai-Jun, Alava, Mikko, Aurell, Erik, and Orponen, Pekka
- Subjects
- *
STATISTICAL sampling , *PROBLEM solving , *ALGORITHMS , *MEAN field theory , *STATISTICAL physics , *PARAMETER estimation - Abstract
The random 3-satisfiability (3-SAT) problem is in the unsatisfiable (UNSAT) phase when the clause density a exceeds a critical value αs ≈ 4.267. Rigorously proving the unsatisfiability of a given large 3-SAT instance is, however, extremely difficult. In this paper we apply the mean-field theory of statistical physics to the unsatisfiability problem, and show that a reduction to 3-XORSAT, which permits the construction of a specific type of UNSAT witnesses (Feige-Kim-Ofek witnesses), is possible when the clause density α > 19. We then construct Feige-Kim-Ofek witnesses for single 3-SAT instances through a simple random sampling algorithm and a focused local search algorithm. The random sampling algorithm works only when a scales at least linearly with the variable number N, but the focused local search algorithm works for clause density a > cNb with b≈0.59 and prefactor c ≈ 8. The exponent b can be further decreased by enlarging the single parameter 5 of the focused local search algorithm. [ABSTRACT FROM AUTHOR]
- Published
- 2013
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