CNF-Satisfiability Test by Counting and Polynomial Average Time

Abstract

The average-case performance of an algorithm for CNF SAT, recently introduced by the author, is discussed. It is shown that the algorithm takes polynomial average time for a class of CNF equations satisfying the condition that for, a constant c, $p^2 v \geqq \ln t - c$, where v is the number of variables, t is the number of clauses, and p is the probability that a given literal appears in a clause. It was known that backtracking plus the pure literal rule, a common way of solving CNF SAT, takes polynomial average time if $p \geqq \varepsilon $ (any small constant) or $p \leqq c(\ln {v / v})^{{3 / 2}} $, but no algorithms were known to take polynomial average time (for all t) in the range $c(\ln v/v)^{3/2} < p < \varepsilon $. For reasonable $t(t \leqq v^\alpha $ for some constant $\alpha > 0$ the new algorithm runs in polynomial average time for $p > (\alpha \ln {v / v})^{{1 / 2}} $, so the unfavorable region is reduced to $c(\ln {v / v})^{{3 / 2}} < p < (\alpha \ln {v / v})^{{1 / 2}} $.