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^{Õ((log log n)2)} time algorithm that computes an (additive) ε approximation to the fraction of satisfying assignments of f for ε = 1/poly(logn). The previous best algorithm due to Luby and Velickovic from nearly two decades ago had a run-time of n^{exp(O(√log log n))}. A crucial ingredient in our algorithm is a structural result which allows us to sparsify any small-width DNFformula. It says that any width w DNF(irrespective of the number of terms) can be ε-approximated by a width w DNFwith at most (w log(1/ε))^O(w) terms. Further, our approximating DNFs 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.