The Complexity of Satisfiability of Small Depth Circuits

Abstract

Say that an algorithm solving a Boolean satisfiability problem x on n variables is improved if it takes time poly(|x|)2^cn for some constant c < 1, i.e., if it is exponentially better than a brute force search. We show an improved randomized algorithm for the satisfiability problem for circuits of constant depth d and a linear number of gates cn: for each d and c, the running time is 2^{(1 − δ)n} where the improvement $\delta\geq 1/O(c^{2^{d-2}-1}\lg^{3\cdot 2^{d-2}-2}c)$, and the constant in the big-Oh depends only on d. The algorithm can be adjusted for use with Grover’s algorithm to achieve a run time of $2^{\frac{1-\delta}{2}n}$ on a quantum computer.