Elad Haramaty

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In this paper we study the structure of polynomials of degree three and four that have high bias or high Gowers norm, over arbitrary prime fields. In particular we obtain the following results. 1. We give a canonical representation for degree three or four polynomials that have a significant bias (i.e. they are not equidistributed). This result generalizes(More)
We consider the problem of testing if a given function $f : \F_q^n \right arrow \F_q$ is close to a $n$-variate degree $d$ polynomial over the finite field $\F_q$ of $q$elements. The natural, low-query, test for this property would be to pick the smallest dimension $t = t_{q,d}\approx d/q$ such that every function of degree greater than $d$reveals this(More)
An r-simple k-path is a path in the graph of length k that passes through each vertex at most r times. The r-SIMPLE k-PATH problem, given a graph G as input, asks whether there exists an r-simple k-path in G. We first show that this problem is NP-Complete. We then show that there is a graph G that contains an r-simple k-path and no simple path of length(More)
In this work we present a strong analysis of the testability of a broad, and to date the most interesting known, class of “affine-invariant” codes. Affine-invariant codes are codes whose coordinates are associated with a vector space and are invariant under affine transformations of the coordinate space. Affine-invariant linear codes form a natural(More)
Let D be a b-wise independent distribution over {0, 1}m. Let E be the “noise” distribution over {0, 1}m where the bits are independent and each bit is 1 with probability η/2. We study which tests f : {0, 1}m → [−1, 1] are ε-fooled by D + E, i.e., |E[f(D+E)]−E[f(U)]| ≤ ε where U is the uniform distribution. We show that D+E ε-fools product tests f : ({0,(More)
A local tester for a code probabilistically views a small set of coordinates of a given word and based on this local view accepts code words with probability one while rejecting words far from the code with constant probability. A local tester for a code is said to be "robust" if the local views of the tester are far from acceptable views when the word(More)
We study a variant of the <i>generalized assignment problem</i> (<scp>gap</scp>), which we label <i>all-or-nothing</i> <scp>gap</scp> (<scp>agap</scp>). We are given a set of items, partitioned into <i>n</i> groups, and a set of <i>m</i> bins. Each item &ell; has size <i>s</i><sub>&ell;</sub> &gt; 0, and utility <i>a</i><sub>&ell;<i>j</i></sub> &ges; 0 if(More)
In the correlated sampling problem, two players, say Alice and Bob, are given two distributions, say P and Q respectively, over the same universe and access to shared randomness. The two players are required to output two elements, without any interaction, sampled according to their respective distributions, while trying to minimize the probability that(More)