Witness of unsatisfiability for a random 3-satisfiability formula

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The random 3-satisfiability (3-SAT) problem is in the unsatisfiable (UNSAT) phase when the clause density α 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 α scales at least linearly with the variable number N, but the focused local search algorithm works for clause density α>cNb with b≈0.59 and prefactor c≈8. The exponent b can be further decreased by enlarging the single parameter S of the focused local search algorithm.

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AlkuperäiskieliEnglanti
Artikkeli052807
Sivut1-10
JulkaisuPhysical Review E
Vuosikerta87
Numero5
TilaJulkaistu - 2013
OKM-julkaisutyyppiA1 Julkaistu artikkeli, soviteltu

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