DNF sparsification and a faster deterministic counting algorithm

Abstract

Given a DNF formula f on n variables, the two natural size measures are the number of terms or size s(f) and the maximum width of a term w(f). It is folklore that small DNF formulas can be made narrow: if a formula has m terms, it can be $${\epsilon}$$ -approximated by a formula with width $${{\rm log}(m/{\epsilon})}$$ . We prove a converse, showing that narrow formulas can be sparsified. More precisely, any width w DNF irrespective of its size can be $${\epsilon}$$ -approximated by a width w DNF with at most $${(w\, {\rm log}(1/{\epsilon}))^{O(w)}}$$ terms. We combine our sparsification result with the work of Luby & Velickovic (1991, Algorithmica 16(4/5):415–433, 1996) to give a faster deterministic algorithm for approximately counting the number of satisfying solutions to a DNF. Given a formula on n variables with poly(n) terms, we give a deterministic $${n^{\tilde{O}({\rm log}\, {\rm log} (n))}}$$ time algorithm that computes an additive $${\epsilon}$$ approximation to the fraction of satisfying assignments of f for $${\epsilon = 1/{\rm poly}({\rm log}\, n)}$$ . The previous best result due to Luby and Velickovic from nearly two decades ago had a run time of $${n^{{\rm exp}(O(\sqrt{{\rm log}\, {\rm log} n}))}}$$ (Luby & Velickovic 1991, in Algorithmica 16(4/5):415–433, 1996).