An Improved Deterministic #SAT Algorithm for Small de Morgan Formulas

Abstract

We give a deterministic #SAT algorithm for de Morgan formulas of size up to $$n^{2.63}$$ , which runs in time $$2^{n-n^{{\varOmega }(1)}}$$ . This improves upon the deterministic #SAT algorithm of Chen et al. (Proceedings of the twenty-ninth annual IEEE conference on computational complexity, 2014), which has similar running time but works only for formulas of size less than $$n^{2.5}$$ . Our new algorithm is based on the shrinkage of de Morgan formulas under random restrictions, shown by Paterson and Zwick (Random Struct Algorithms 4(2):135–150, 1993). We prove a concentrated and constructive version of their shrinkage result. Namely, we give a deterministic polynomial-time algorithm that selects variables in a given de Morgan formula so that, with high probability over the random assignments to the chosen variables, the original formula shrinks in size, when simplified using a given deterministic polynomial-time formula-simplification algorithm.