The paper investigates expansion properties of the Grassmann graph,
motivated by recent results of [KMS, DKKMS] concerning hardness
of the Vertex-Cover and of the $2$-to-$1$ Games problems. Proving the
hypotheses put forward by these papers seems to first require a better
understanding of these expansion properties.

We consider the edge ... more >>>

We propose a combinatorial hypothesis regarding a subspace vs. subspace agreement test, and prove that if correct it leads to a proof of the 2-to-1 Games Conjecture, albeit with imperfect completeness.

The first part of this thesis strengthens the low-error PCP
characterization of NP, coming closer to the upper limit of the
conjecture of~\cite{BGLR}.

In the second part we show that a boolean function over
$n$ variables can be tested for the property of depending ... more >>>

We show that every language in NP has a PCP verifier that tosses $O(\log n)$ random coins, has perfect completeness, and a soundness error of at most $1/poly(n)$, while making at most $O(poly\log\log n)$ queries into a proof over an alphabet of size at most $n^{1/poly\log\log n}$. Previous constructions that ... more >>>

For a string $a \in \{0,1\}^n$ its $k$-fold direct sum encoding is a function $f_a$ that takes as input sets $S \subseteq [n]$ of
size $k$ and outputs $f_a(S) = \sum_{i \in S} a_i$.
In this paper we are interested in the Direct Sum Testing Problem,
where we are given ... more >>>

We present new explicit constructions of *deterministic* randomness extractors, dispersers and related objects. We say that a
distribution $X$ on binary strings of length $n$ is a
$\delta$-source if $X$ assigns probability at most $2^{-\delta n}$
to any string of length $n$. For every $\delta>0$ we construct the
following poly($n$)-time ... more >>>

Motivated by the study of Parallel Repetition and also by the Unique
Games Conjecture, we investigate the value of the ``Odd Cycle Games''
under parallel repetition. Using tools from discrete harmonic
analysis, we show that after $d$ rounds on the cycle of length $m$,
the value of the game is ... more >>>

In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of $\GW + \eps$, for all $\eps > 0$; here $\GW \approx .878567$ denotes the approximation ratio achieved by the Goemans-Williamson algorithm~\cite{GW95}. This implies that if the Unique ... more >>>

Optimal dispersers have better dependence on the error than
optimal extractors. In this paper we give explicit disperser
constructions that beat the best possible extractors in some
parameters. Our constructions are not strong, but we show that
having such explicit strong constructions implies a solution
to the Ramsey graph construction ... more >>>

This paper studies the computational complexity of the following type of
quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find $x \in \{-1,+1\}^n$ that maximizes $x^TA x$. This problem recently attracted attention due to its application in various clustering settings (Charikar and Wirth, 2004) as well as ... more >>>

This paper strengthens the low-error PCP characterization of NP, coming
closer to the ultimate BGLR conjecture. Namely, we prove that witnesses for
membership in any NP language can be verified with a constant
number of accesses, and with an error probability exponentially
small in the ... more >>>

This paper shows finding the closest vector in a lattice
to be NP-hard to approximate to within any factor up to
$2^{(\log{n})^{1-\epsilon}}$ where
$\epsilon = (\log\log{n})^{-\alpha}$
and $\alpha$ is any positive constant $<{1\over 2}$.