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).
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Gopalan, P., Meka, R. & Reingold, O. DNF sparsification and a faster deterministic counting algorithm. comput. complex. 22, 275–310 (2013). https://doi.org/10.1007/s00037-013-0068-6
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DOI: https://doi.org/10.1007/s00037-013-0068-6