Ido Ben-Eliezer

Comparing the strength of query types in property testing: the case of testing k -colorability

We study the power of four query models in the context of property testing in general graphs, where our main case study is the problem of testing k-colorability. Two query types, which have been studied extensively in the past, are pair queries and neighbor queries. The former corresponds to asking whether there is an edge between any particular pair of vertices, and the latter to asking for the ith neighbor of a particular vertex. We show that while for pair queries testing k-colorability requires a number of queries that is a monotone decreasing function in the average degree d, the query complexity in the case of neighbor queries remains roughly the same for every density and for large values of k. We also consider a combined model that allows both types of queries, and we propose a new, stronger, query model, related to the field of Group Testing. We give upper and lower bounds on the query complexity for one-sided error in all the models, where the bounds are nearly tight for three of the models. In some of the cases, our lower bounds extend to two-sided error algorithms.The problem of testing k-colorability was previously studied in the contexts of dense graphs and of sparse graphs, and in our proofs we unify approaches from those cases, and also provide some new tools and techniques that may be of independent interest.

Long cycles in subgraphs of (pseudo)random directed graphs

We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every 08γ81/2 we find a constant c = c(γ) such that the following holds. Let G = (V, E) be a (pseudo)random directed graph on n vertices and with at least a linear number of edges, and let G′ be a subgraph of G with (1/2 + γ)|E| edges. Then G′ contains a directed cycle of length at least (c − o(1))n. Moreover, there is a subgraph G′′of G with (1/2 + γ − o(1))|E| edges that does not contain a cycle of length at least cn.

Random Low Degree Polynomials are Hard to Approximate

We study the problem of how well a typical multivariate polynomial can be approximated by lower degree polynomials over

Small Sample Spaces Cannot Fool Low Degree Polynomials

{\mathbb{F}_{2}}

Random low-degree polynomials are hard to approximate

. We prove that, with very high probability, a random degree d + 1 polynomial has only an exponentially small correlation with all polynomials of degree d , for all degrees d up to

The size Ramsey number of a directed path

\Theta\left(n\right)

Polynomial Threshold Functions: Structure, Approximation and Pseudorandomness

. That is, a random degree d + 1 polynomial does not admit a good approximation of lower degree. In order to prove this, we prove far tail estimates on the distribution of the bias of a random low degree polynomial. Recently, several results regarding the weight distribution of Reed---Muller codes were obtained. Our results can be interpreted as a new large deviation bound on the weight distribution of Reed---Muller codes.

Biased orientation games

A distribution Don a set

Perfectly Balanced Partitions of Smoothed Graphs

S \subset \mathbb Z_{p}^{N}

Random low degree polynomials are hard to approximate.

i¾?-fools polynomials of degree at most din Nvariables over