DNF Sparsification and a Faster Deterministic Counting Algorithm [article]

Parikshit Gopala, Raghu Meka, Omer Reingold
2012 arXiv   pre-print
Given a DNF formula 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 short DNF formulas can be made narrow. We prove a converse, showing that narrow formulas can be sparsified. More precisely, any width w DNF irrespective of its size can be ϵ-approximated by a width w DNF with at most (w(1/ϵ))^O(w) terms. We combine our sparsification result with the work of Luby and Velikovic to give a faster
more » ... tic 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^Õ((n)) time algorithm that computes an additive ϵ approximation to the fraction of satisfying assignments of f for ϵ = 1/( n). The previous best result due to Luby and Velickovic from nearly two decades ago had a run-time of n^(O(√( n))).
arXiv:1205.3534v1 fatcat:e23ypu6yafgzni35gvz6eqzqgi