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Focused random walk (FRW) is one of the most influential paradigm of stochastic local search (SLS) algorithms for the propositional satisfiability (SAT) problem. Recently, an interesting probability distribution (PD) strategy for variable selection was proposed and has been successfully used to improve SLS algorithms, resulting in state-of-the-art solvers. However, most solvers based on the PD strategy only use polynomial function (PoF) to handle the exponential decay and are still unsatisfactory in dealing with medium and huge k -SAT instances at and near the phase transition. The present paper is focused on handling all k -SAT instances with long clauses. Firstly, an extensive empirical study of one state-of-the-art FRW solver WalkSATlm on a wide range of SAT problems is presented with the focus given on fitting the distribution of the break value of variable selected in each step, which turns out to be a Boltzmann function. Using theses case studies as a basis, we propose a pseudo normal function (PNF) to fit the distribution of the break value of variable selected, which is actually a variation of the Boltzmann function. In addition, a new tie-breaking flipping (TBF) strategy is proposed to prevent the same variable from being flipped in consecutive steps. The PNF based PD strategy combined with the TBF strategy lead to a new variable selection heuristic named PNF-TBF. The PNF-TBF heuristic along with a variable allocation value (Vav) function are used to significantly improve ProbSAT, a state-of-the-art SLS solver, leads to a new FRW algorithm dubbed PNFSat, which achieves the state-of-the-art performance on a broad range of huge random 7-SAT instance near the phase transition as demonstrated via the extensive experimental studies. Some further improved versions on top of PNFSat are presented respectively, including PNFSat_alt, which achieves the state-of-the-art performance on the medium 7-SAT instances at the phase transition; PN&PoFSat, which achieves the state-of-the-art performance on a broad range of random 5-SAT benchmarks; as well as an integrated version of these three algorithms, named PDSat, which achieves the state-of-the-art performances on all huge and medium random k -SAT instances with long clauses as demonstrated via the comparative studies using different benchmarks.

Focused random walk with probability distribution for SAT with long clauses