  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 $\epsilon$-approximated by a width $w$ DNF with
at most $(w\log(1/\epsilon))^{O(w)}$ terms.
  We combine our sparsification result with the work of Luby and Velikovic 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 $n^{\tilde{O}(\log \log(n))}$ 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
result due to Luby and Velickovic from nearly two decades ago had a run-time of
$n^{\exp(O(\sqrt{\log \log n}))}$.
