We give new constructions of randomness extractors and lossless condensers that are optimal to within constant factors in both the seed length and the output length. For extractors, this matches the parameters of the current best known construction [LRVW03]; for lossless condensers, the previous best constructions achieved optimality to within ... more >>>

The importance of {\em width} as a resource in resolution theorem proving
has been emphasized in work of Ben-Sasson and Wigderson. Their results show that lower
bounds on the size of resolution refutations can be proved in a uniform manner by
demonstrating lower bounds on the width ... more >>>

We study linear programming relaxations of Vertex Cover and Max Cut
arising from repeated applications of the ``lift-and-project''
method of Lovasz and Schrijver starting from the standard linear
programming relaxation.

For Vertex Cover, Arora, Bollobas, Lovasz and Tourlakis prove that
the integrality gap remains at least $2-\epsilon$ after
$\Omega_\epsilon(\log n)$ ... more >>>