A Note on Approximate Counting for k-DNF

Abstract

We describe a deterministic algorithm that, for constant k, given a k-DNF or k-CNF formula φ and a parameter ε, runs in time linear in the size of φ and polynomial in 1/ε (but doubly exponential in k) and returns an estimate of the fraction of satisfying assignments for φ up to an additive error ε. This improves over previous polynomial (but super-linear) time algorithms. The algorithm uses a simple recursive procedure and it is not based on derandomization techniques. It is similar to an algorithm by Hirsch for the related problem of solving k-SAT under the promise that an ε-fraction of the assignments are satisfying. Our analysis is different from (and somewhat simpler than) Hirsch’s.

We also note that the argument that we use in the analysis of the algorithm gives a proof of a result of Luby and Velickovic that every k-CNF is “fooled” by every δ-biased distribution, with $\delta = 1/2^{O(k2^k)}$.