Solving Sparse Instances of Max SAT via Width Reduction and Greedy Restriction

Abstract

We present a moderately exponential time and polynomial space algorithm for sparse instances of Max SAT. Our algorithms run in time of the form O2(1?μ(c))n$O\left (2^{(1-\mu (c))n}\right )$ for instances with n variables and cn clauses. Our deterministic and randomized algorithm achieve μ(c)=Ω1c2log2c$\mu (c) = {\Omega }\left (\frac {1}{c^{2}\log ^{2} c}\right )$ and μ(c)=Ω1clog3c$\mu (c) = {\Omega }\left (\frac {1}{c \log ^{3} c}\right )$ respectively. Previously, an exponential space deterministic algorithm with μ(c)=Ω1clogc$\mu (c) = {\Omega }\left (\frac {1}{c\log c}\right )$ was shown by Dantsin and Wolpert [SAT 2006] and a polynomial space deterministic algorithm with μ(c)=Ω12O(c)$\mu (c) = {\Omega }\left (\frac {1}{2^{O(c)}}\right )$ was shown by Kulikov and Kutzkov [CSR [2007]]. Our algorithms have three new features. They can handle instances with (1) weights and (2) hard constraints, and also (3) they can solve counting versions of Max SAT. Our deterministic algorithm is based on the combination of two techniques, width reduction of Schuler and greedy restriction of Santhanam. Our randomized algorithm uses random restriction instead of greedy restriction.