We present a deterministic algorithm that counts the number of satisfying assignments for any de Morgan formula F of size at most n3−16ε in time 2n−n ε · poly(n), for any small constant ε > 0. We do this by derandomizing the randomized algorithm mentioned by Komargodski et al. (FOCS, 2013) and Chen et al. (CCC, 2014). Our result uses the tight “shrinkage in expectation” result of de Morgan form...

This paper analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems, without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP 6⊆ coNP/poly, no efficient preprocessing algorithm can reduce n-variable instances of cnf-sat with d literals per clause, to equiv...

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 ε > 0, we show the existence of an n v...

Carbon nanotube (CNT)-based materials allow for the direct conversion of light into heat and then mechanical force in so-called photo-mechanical effect. This effect has been observed almost solely form polymer (nano)composites, where CNTs act as active fillers. To overcome issue with heterogeneous distribution, hybrids based on multiwalled carbon nanotubes (MWCNTs) covalently modified poly(2-hy...

We study k-SVD that is to obtain the first k singular vectors of a matrix A approximately. Recently, a few breakthroughs have been discovered on k-SVD: Musco and Musco [18] provided the first gap-free theorem for the block Krylov method, Shamir [20] discovered the first variance-reduction stochastic method, and Bhojanapalli et al. [6] provided the fastest O(nnz(A) + poly(1/ε))-type of algorithm...

We study k-SVD that is to obtain the first k singular vectors of a matrix A. Recently, a few breakthroughs have been discovered on k-SVD: Musco and Musco [19] proved the first gap-free convergence result using the block Krylov method, Shamir [21] discovered the first variance-reduction stochastic method, and Bhojanapalli et al. [7] provided the fastestO(nnz(A)+ poly(1/ε))-time algorithm using a...

The heteronucleobase (adenine and uracil)-functionalized poly(ε-caprolactone) (A-PCL-U) possessing supramolecular structure has been successfully synthesized through the combination of ringopening polymerization and Michael addition reaction. Attachment of multiple hydrogen-bonding units to chain ends of PCL results in phase separation and substantial increase in the viscosity. The association ...