Assaf Naor

Approximating the cut-norm via Grothendieck's inequality

The <i>cut-norm</i> ||A||<inf>C</inf> of a real matrix A=(a<inf>ij</inf>)<inf>i∈ R,j∈S</inf> is the maximum, over all I ⊂ R, J ⊂ S of the quantity | Σ<inf>i ∈ I, j ∈ J</inf> a<inf>ij</inf>|. This concept plays a major role in the design of efficient approximation algorithms for dense graph and matrix problems. Here we show that the problem of approximating the cut-norm of a given real matrix is MAX SNP hard, and provide an efficient approximation algorithm. This algorithm finds, for a given matrix A=(a<inf>ij</inf>)<inf>i ∈ R, j ∈ S</inf>, two subsets I ⊂ R and J ⊂ S, such that | Σ<inf>i ∈ I, j ∈ J</inf> a<inf>ij</inf>| ≥ ρ ||A||<inf>C</inf>, where ρ > 0 is an absolute constant satisfying

Rigorous location of phase transitions in hard optimization problems

ρ > 0. 56. The algorithm combines semidefinite programming with a rounding technique based on Grothendiecks Inequality. We present three known proofs of Grothendiecks inequality, with the necessary modifications which emphasize their algorithmic aspects. These proofs contain rounding techniques which go beyond the random hyperplane rounding of Goemans and Williamson [12], allowing us to transfer various algorithms for dense graph and matrix problems to the sparse case.

Solution of Shannon's problem on the monotonicity of entropy

It is widely believed that for many optimization problems, no algorithm is substantially more efficient than exhaustive search. This means that finding optimal solutions for many practical problems is completely beyond any current or projected computational capacity. To understand the origin of this extreme ‘hardness’, computer scientists, mathematicians and physicists have been investigating for two decades a connection between computational complexity and phase transitions in random instances of constraint satisfaction problems. Here we present a mathematically rigorous method for locating such phase transitions. Our method works by analysing the distribution of distances between pairs of solutions as constraints are added. By identifying critical behaviour in the evolution of this distribution, we can pinpoint the threshold location for a number of problems, including the two most-studied ones: random k-SAT and random graph colouring. Our results prove that the heuristic predictions of statistical physics in this context are essentially correct. Moreover, we establish that random instances of constraint satisfaction problems have solutions well beyond the reach of any analysed algorithm.

Nearest-neighbor-preserving embeddings

It is shown that if X1, X2, . . . are independent and identically distributed square-integrable random variables then the entropy of the normalized sum Ent (X1+ · · · + Xn over √n) is an increasing function of n. This resolves an old problem which goes back to [6, 7, 5]. The result also has a version for non-identically distributed random variables or random vectors.

Embedding the diamond graph in L p and dimension reduction in L 1

In this article we introduce the notion of nearest-neighbor-preserving embeddings. These are randomized embeddings between two metric spaces which preserve the (approximate) nearest-neighbors. We give two examples of such embeddings for Euclidean metrics with low “intrinsic” dimension. Combining the embeddings with known data structures yields the best-known approximate nearest-neighbor data structures for such metrics.

Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces

AbstractWe show that any embedding of the level k diamond graph of Newman and Rabinovich [NR] into Lp, 1 < p ≤ 2, requires distortion at least

Boolean functions whose Fourier transform is concentrated on the first two levels

\sqrt{k(p-1) + 1}

Lp metrics on the Heisenberg group and the Goemans-Linial conjecture

. An immediate corollary is that there exist arbitrarily large n-point sets