An Improved Deterministic #SAT Algorithm for Small De Morgan Formulas.
Ruiwen Chen, Valentine Kabanets, Nitin Saurabh · Electronic colloquium on computational complexity · 2013
We give a deterministic #SAT algorithm for de Morgan formulas of size up to $$n^{2.63}$$n2.63, which runs in time $$2^{n-n^{{\varOmega }(1)}}$$2n-nΩ(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 $$n^{2.5}$$n2.5. 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.