18,558 research outputs found

    DNF Sparsification and a Faster Deterministic Counting Algorithm

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    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 ϡ\epsilon-approximated by a width ww DNF with at most (wlog⁑(1/ϡ))O(w)(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 nO~(log⁑log⁑(n))n^{\tilde{O}(\log \log(n))} time algorithm that computes an additive ϡ\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 nexp⁑(O(log⁑log⁑n))n^{\exp(O(\sqrt{\log \log n}))}.Comment: To appear in the IEEE Conference on Computational Complexity, 201

    Subspace-Invariant AC0^0 Formulas

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    We consider the action of a linear subspace UU of {0,1}n\{0,1\}^n on the set of AC0^0 formulas with inputs labeled by literals in the set {X1,Xβ€Ύ1,…,Xn,Xβ€Ύn}\{X_1,\overline X_1,\dots,X_n,\overline X_n\}, where an element u∈Uu \in U acts on formulas by transposing the iith pair of literals for all i∈[n]i \in [n] such that ui=1u_i=1. A formula is {\em UU-invariant} if it is fixed by this action. For example, there is a well-known recursive construction of depth d+1d+1 formulas of size O(nβ‹…2dn1/d)O(n{\cdot}2^{dn^{1/d}}) computing the nn-variable PARITY function; these formulas are easily seen to be PP-invariant where PP is the subspace of even-weight elements of {0,1}n\{0,1\}^n. In this paper we establish a nearly matching 2d(n1/dβˆ’1)2^{d(n^{1/d}-1)} lower bound on the PP-invariant depth d+1d+1 formula size of PARITY. Quantitatively this improves the best known Ξ©(2184d(n1/dβˆ’1))\Omega(2^{\frac{1}{84}d(n^{1/d}-1)}) lower bound for {\em unrestricted} depth d+1d+1 formulas, while avoiding the use of the switching lemma. More generally, for any linear subspaces UβŠ‚VU \subset V, we show that if a Boolean function is UU-invariant and non-constant over VV, then its UU-invariant depth d+1d+1 formula size is at least 2d(m1/dβˆ’1)2^{d(m^{1/d}-1)} where mm is the minimum Hamming weight of a vector in UβŠ₯βˆ–VβŠ₯U^\bot \setminus V^\bot
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