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We give a deterministic #SAT algorithm for de Morgan formulas of size up to n 2.63 , which runs in time 2 n−n Ω(1) . This improves upon the deterministic #SAT algorithm of [3], which has similar running time but works only for formulas of size less than n 2.5 . Our new algorithm is based on the shrinkage of de Morgan formulas under random restrictions, shown by Paterson and Zwick [12]. We prove a concentrated and constructive version of their shrinkage result. Namely, we give a deterministic polynomial-time algorithm that selects variables in a given de Morgan formula so that, with high probability over the random assignments to the chosen variables, the original formula shrinks in size, when simplified using a deterministic polynomial-time formulasimplification algorithm.

Abstract. We provide a list of new natural VNP-intermediate polynomial families, based on basic (combinatorial) NP-complete problems that are complete under parsimonious reductions. Over finite fields, these families are in VNP, and under the plausible hypothesis ModpP ⊆ P/poly, are neither VNP-hard (even under oracle-circuit reductions) nor in VP. Prior to this, only the Cut Enumerator polynomial was known to be VNP-intermediate, as shown by Bürgisser in 2000. We next show that over rationals and reals, two of our intermediate polynomials, based on satisfiability and Hamiltonian cycle, are not monotone affine polynomial-size projections of the permanent. This augments recent results along this line due to Grochow. Finally, we describe a (somewhat natural) polynomial defined independent of a computation model, and show that it is VP-complete under polynomial-size projections. This complements a recent result of Durand et al. (2014) which established VP-completeness of a related polynomial but under constant-depth oracle circuit reductions. Both polynomials are based on graph homomorphisms. A simple restriction yields a family similarly complete for VBP.

The Fourier-Entropy Influence (FEI) Conjecture states that for any Boolean function f : {+1, −1} n → {+1, −1}, the Fourier entropy of f is at most its influence up to a universal constant factor. While the FEI conjecture has been proved for many classes of Boolean functions, it is still not known whether it holds for the class of Linear Threshold Functions. A natural question is: Does the FEI conjecture hold for a "random" linear threshold function? In this paper, we answer this question in the affirmative. We consider two natural distributions on the weights defining a linear threshold function, namely uniform distribution on [−1, 1] and Normal distribution.

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