On Computing k-CNF Formula Properties

Abstract

The latest generation of SAT solvers (e.g. [10,7]) generally have three key features: randomization of variable selection, backtracking search, and some form of clause learning. We present a simple algorithm with these three features and prove that for instances with constant Δ (where Δ is the clause-to-variable ratio) the algorithm indeed has good worst-case performance, not only for computing SAT/UNSAT but more general properties as well, such as maximum satisfiability and counting the number of satisfying assignments. In general, the algorithm can determine any property that is computable via self-reductions on the formula.

One corollary of our findings is that for all fixed Δ and k ≥ 2, Max-k -SAT is solvable in O(c ⁿ) expected time for some c < 2, partially resolving a long-standing open problem in improved exponential time algorithms. For example, when Δ = 4.2 and k = 3, Max-k -SAT is solvable in O(1.8932ⁿ) worst-case expected time. We also improve the known time bounds for exact solution of #2SAT and #3SAT, and the bounds for k-SAT when k ≥ 5.