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  • Wu, Lu-Lu, et al. (author)
  • Witness of unsatisfiability for a random 3-satisfiability formula
  • 2013
  • In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics. - 1539-3755 .- 1550-2376. ; 87:5, s. 052807-
  • Journal article (peer-reviewed)abstract
    • The random 3-satisfiability (3-SAT) problem is in the unsatisfiable (UNSAT) phase when the clause density alpha exceeds a critical value alpha(s) approximate to 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 alpha > 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 alpha > cN(b) with b approximate to 0.59 and prefactor c approximate to 8. The exponent b can be further decreased by enlarging the single parameter S of the focused local search algorithm.
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