Witness of unsatisfiability for a random 3-satisfiability formula

Lu-Lu Wu, Hai-Jun Zhou, Mikko Alava, Erik Aurell, Pekka Orponen

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Abstract

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.
Original languageEnglish
Article number052807
Pages (from-to)1-10
JournalPhysical Review E
Volume87
Issue number5
DOIs
Publication statusPublished - 2013
MoE publication typeA1 Journal article-refereed

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