An Approximation Algorithm for #k-SAT

TL;DR

This paper introduces a randomized approximation algorithm for counting solutions to #k-SAT problems, achieving better runtime than exact algorithms for k >= 3, with controllable bounds and error tolerance.

Contribution

It presents the first efficient randomized approximation scheme for #k-SAT with polynomial dependence on error and mildly exponential dependence on variables.

Findings

Approximate #k-SAT solutions in sub-exponential time for k >= 3.

Provides bounds on the number of solutions with controllable accuracy.

Achieves faster runtimes compared to exact algorithms for large instances.

Abstract

We present a simple randomized algorithm that approximates the number of satisfying assignments of Boolean formulas in conjunctive normal form. To the best of our knowledge this is the first algorithm which approximates #k-SAT for any k >= 3 within a running time that is not only non-trivial, but also significantly better than that of the currently fastest exact algorithms for the problem. More precisely, our algorithm is a randomized approximation scheme whose running time depends polynomially on the error tolerance and is mildly exponential in the number n of variables of the input formula. For example, even stipulating sub-exponentially small error tolerance, the number of solutions to 3-CNF input formulas can be approximated in time O(1.5366^n). For 4-CNF input the bound increases to O(1.6155^n). We further show how to obtain upper and lower bounds on the number of solutions to a CNF formula in a controllable way. Relaxing the requirements on the quality of the approximation, on k-CNF input we obtain significantly reduced running times in comparison to the above bounds.