An Improved Deterministic #SAT Algorithm for Small De Morgan Formulas

Ruiwen Chen, Valentine Kabanets, and Nitin Saurabh


We give a deterministic #SAT algorithm for de Morgan formulas of size up to $n^{2.63}$, which runs in time $2^{n-n^{\Omega(1)}}$. This improves upon the deterministic #SAT algorithm of Chen et al., 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. 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 deterministic polynomial-time formula-simplification algorithm.