The Pure Literal Rule and Polynomial Average Time

Abstract

For a simple parameterized model of conjunctive normal form predicates, we show that a simplified version of the Davis–Putnam procedure can, for many values of the parameters, solve the satisfiability problem in polynomial average time. Let v be the number of variables, $t(v)$ the number of clauses in a predicate, and $p(v)$ the probability that a given literal appears in a clause ($p(v)$ is the same for all literals). Let $\varepsilon $ be any small positive constant and n any large positive integer. Then a version of the Davis–Putnam procedure that uses only backtracking and the pure literal rule uses average time that is polynomial in the problem size when any of the following conditions are true for large v. (1) $t(v) \leqq n\ln v$; (2) $t(v) \geqq \exp (\varepsilon v)$; (3) $p(v) \geqq \varepsilon $; or (4) $p(v) \leqq n (\ln v/ v)^{3/2} $. Until recently the best previous bounds for cases (1) and (4) were $t(v) \leqq (\ln\ln v)/(\ln3)$ and $p(v) \leqq \exp ( - v\ln \ln v)$. These results show that the problem types for which the pure literal rule works well are quite different from those for which backtracking works well. Our present knowledge suggests this random problem sets with $t(v)$ somewhat larger than v and with $p(v)$ somewhat larger than $v^{ - 1} $ are particularly difficult to solve.