Ishay Haviv

Tensor-based hardness of the shortest vector problem to within almost polynomial factors

We show that unless NP ⊆ RTIME (2^{poly(log n)}), for any ε > 0 there is no polynomial-time algorithm approximating the Shortest Vector Problem (SVP) on <i>n</i>-dimensional lattices inthe l_p norm (1 ≤q p<∞) to within a factor of 2^{(log n)^1-ε}. This improves the previous best factor of 2^{(logn)^1/2-ε} under the same complexity assumption due to Khot. Under the stronger assumption NP ࣰ RSUBEXP, we obtain a hardness factor of n^{c/log log n} for some c > 0. Our proof starts with Khots SVP instances from that are hard to approximate to within some constant. To boost the hardness factor we simply apply the standard tensor product oflattices. The main novel part is in the analysis, where we show that Khots lattices behave nicely under tensorization. At the heart of the analysis is a certain matrix inequality which was first used in the context of lattices by de Shalit and Parzanchevski.

On linear index coding for random graphs

In the index coding problem, the goal is to transmit an n character word over a field F to n receivers (one character per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword broadcasted to all receivers which allows each receiver to learn its character. For linear index coding, the minimum possible length is known to be equal to the minrank parameter. In this paper we initiate the study of the typical minimum length of a linear index code for the random graph G(n, p) over a field F. First, we prove that for every constant size field F and a constant p, the minimum length of a linear index code for G(n, p) over F is almost surely Ω(√n). Second, we introduce and study two special models of index coding and study their typical minimum length: Locally decodable index codes in which the receivers are required to query at most q characters from the encoded message (such codes naturally correspond to efficient decoding); and low density index codes in which every character of the broadcasted word affects at most q characters in the encoded message (such codes naturally correspond to efficient encoding procedures). We present enhanced results for these special models.

The restricted isometry property of subsampled fourier matrices

A matrix A ∈ Cq×N satisfies the restricted isometry property of order k with constant e if it preserves the e2 norm of all k-sparse vectors up to a factor of 1 ± e. We prove that a matrix A obtained by randomly sampling q = O(k · log2 k · log N) rows from an N ×N Fourier matrix satisfies the restricted isometry property of order k with a fixed e with high probability. This improves on Rudelson and Vershynin (Comm. Pure Appl. Math., 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).

Hardness of the covering radius problem on lattices

We provide the first hardness result for the covering radius problem on lattices (CRP). Namely, we show that for any large enough p les infin there exists a constant cp > 1 such that CRP in the lscrp norm is Pi2-hard to approximate to within any constant less than cp. In particular, for the case p = infin, we obtain the constant Cinfin = 1.5. This gets close to the constant 2 beyond which the problem is not believed to be Pi2-hard. As part of our proof, we establish a stronger hardness of approximation result for the forallexist-3-SAT problem with bounded occurrences. This hardness result might be useful elsewhere

Beating the Gilbert-Varshamov bound for online channels

In the online channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword x = (x1, …, xn) ∈ {0, 1}n bit by bit via a channel limited to at most pn corruptions. The channel is online in the sense that at the ith step the channel decides whether to flip the ith bit or not and its decision is based only on the bits transmitted so far, i.e., (x1, …, xi). This is in contrast to the classical adversarial channel in which the corruption is chosen by a channel that has full knowledge on the sent codeword x. The best known lower bound on the capacity of both the online channel and the classical adversarial channel is the well-known Gilbert-Varshamov bound. In this paper we prove a lower bound on the capacity of the online channel which beats the Gilbert-Varshamov bound for any positive p such that H(2p) &#60; 1 over 2 (where H is the binary entropy function).

On the Hardness of Satisfiability with Bounded Occurrences in the Polynomial-Time Hierarchy

\newcommand{\SVP}{\mathsf{SVP}} \newcommand{\NP}{\mathsf{NP}} \newcommand{\RTIME}{\mathsf{RTIME}} \newcommand{\RSUBEXP}{\mathsf{RSUBEXP}} \newcommand{\eps}{\epsilon} \newcommand{\poly}{\mathop{\mathrm{poly}}}

THE EUCLIDEAN DISTORTION OF FLAT TORI

We show that unless

H-wise independence

\NP \subseteq \RTIME (2^{\poly(\log{n})})

Dioid partitions of groups

, there is no polynomial-time algorithm approximating the Shortest Vector Problem (