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 $(wlog(1/epsilon))^{O(w)}$ 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 $n^{exp(O(sqrt{log log n}))}$.
The long code is a central tool in hardness of approximation, especially in questions related to the unique games conjecture. We construct a new code that is exponentially more efficient, but can still be used in many of these applications. Using the
new code we obtain exponential improvements over several known results, including the following: 1. For any eps > 0, we show the existence of an n vertex graph G where every set of o(n) vertices has expansion 1 - eps, but Gs adjacency matrix has more than exp(log^delta n) eigenvalues larger than 1 - eps, where delta depends only on eps. This answers an open question of Arora, Barak and Steurer (FOCS 2010) who asked whether one can improve over the noise graph on the Boolean hypercube that has poly(log n) such eigenvalues. 2. A gadget that reduces unique games instances with linear constraints modulo K into instances with alphabet k with a blowup of K^polylog(K), improving over the previously known gadget with blowup of 2^K. 3. An n variable integrality gap for Unique Games that that survives exp(poly(log log n)) rounds of the SDP + Sherali Adams hierarchy, improving on the previously known bound of poly(log log n). We show a connection between the local testability of linear codes and small set expansion in certain related Cayley graphs, and use this connection to derandomize the noise graph on the Boolean hypercube.
