We consider the satisfiability problem for circuits of limited size and/or depth. Say that an algorithm solving a Boolean satisfiability problem on n variables is improved iff it takes time O(2 cn ) for some constant c < 1, i.e. iff it is exponentially better than a brute force search. We show an improved algorithm for the satisfiability problem for circuits of constant depth and linear size. If improved upper bounds are not possible for a variant where the size is somewhat more than linear or the depth grows, can we provide evidence regarding the hardness of the problem? (note to authors: Did we actually discuss this question in greater depth in the paper? Maybe this should be reworded.) For each d and c, we give a randomized algorithm solving the satisfiability problem for depth d circuits with n variables and at most cn gates in time 2 (1−δ)n where δ ≥ 1/O(c −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 1−δ 2 n on a quantum computer.
The first is an algorithm due to Calabro, Impagliazzo and Paturi REF which has a running time of the form c n , c < 2 for satisfiability of constant-depth Boolean circuits of linear size.
The Complexity of Satisfiability of Small Depth Circuits
IWPEC
Mathematics Computer Science