Analyses on the 2 and 3-Flip Neighborhoods for the MAX SAT

Mutsunori Yagiura and Toshihide Ibaraki
For problems SAT and MAX SAT, local search algorithms are widely acknowledged as one of the most effective approaches. Most of the local search algorithms are based on the 1-flip neighborhood, which is the set of solutions obtainable by flipping the truth assignment of one variable. In this paper, we consider $r$-flip neighborhoods for $r \geq 2$, and propose, for $r=2, 3$, new implementations that reduce the number of candidates in the neighborhood without sacrificing the solution quality. For 2-flip (resp., 3-flip) neighborhood, we show that its expected size is $O(n+m)$ (resp., $O(m+t^2n)$), which is usually much smaller than the original size $O(n^2)$ (resp., $O(n^3)$), where $n$ is the number of variables, $m$ is the number of clauses and $t$ is the maximum number of appearances of one variable. Computational results tell that these estimates by the expectation well represent the real performance.

Key Words: MAX SAT, SAT, local search, neighborhood

Journal of Combinatorial Optimization, Vol. 3, No. 1, pp. 95-114, July 1999.

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