This work addresses k-restriction problems, which unify combinatorial problems of the following type: The goal is to construct a short list of strings in Î£m that satisfies a given set of k-wiseâ€¦ (More)

We consider the possibility of basing one-way functions on NP-Hardness; that is, we study possible reductions from a worst-case decision problem to the task of average-case inverting aâ€¦ (More)

We establish two results about the inapproximability of the Densest Îº-Subgraph (DÎºS) problem. Both results are of similar flavor: ruling out constant factor approximations in polynomial time for theâ€¦ (More)

We introduce and study a new model of interactive proofs: AM(k), or Arthur-Merlin with k non-communicating Merlins. Unlike with the better-known MIP, here the assumption is that each Merlin receivesâ€¦ (More)

One can learn any hypothesis class H with O(log |H|) labeled examples. Alas, learning with so few examples requires saving the examples in memory, and this requires |X | memory states, where X is theâ€¦ (More)

We show an alternative proof of the Schwartz-Zippel lemma. The lemma bounds the number of zeros of a multivariate low degree polynomial over a finite field.

Given a function f:Fmâ†’F over a finite field F, a low degree tester tests its agreement with an m-variate polynomial of total degree at most d over F. The tester is usually given access to an oracle Aâ€¦ (More)

We propose a candidate reduction for ruling out polynomial-time algorithms for unique games, either under plausible complexity assumptions, or unconditionally for Lasserre semi-definite programs withâ€¦ (More)