DNF Sparsification and a Faster Deterministic Counting Algorithm

We give a faster deterministic algorithm for approximately counting the number of satisfying solutions to a \dnf\ or \cnf. Given a \dnf (or \cnf) $f$ on $n$ variables and $\poly(n)$ terms, we give a deterministic $n^{\tilde{O}((\log \log n)^2)}$ time algorithm that computes an (additive) $\epsilon$ approximation to the fraction of satisfying assignments of $f$ for $\epsilon = 1/\poly(\log n)$. The previous best algorithm due to Luby and Velickovic from nearly two decades ago had a run-time of $n^{\exp(O(\sqrt{\log \log n}))}$. A crucial ingredient in our algorithm is a structural result which allows us to sparsify any small-width \dnf formula. It says that any width $w$ \dnf (irrespective of the number of terms) can be $\epsilon$-approximated by a width $w$ \dnf with at most $(w\log(1/\epsilon))^{O(w)}$ terms. Further, our approximating \dnf s have an additional ``sandwiching'' property which is crucial for applications to derandomization. We believe the sparsification result to be of independent interest and use it to show a weak derandomization of the switching lemma wherein the random restrictions need only have limited independence.