# Computer Science and Discrete Mathematics (CSDM)

## New Independent Source Extractors with Exponential Improvement

## Sparsity Lower Bounds for Dimensionality Reducing Maps

Abstract:

We give near-tight lower bounds for the sparsity required in several dimensionality reducing linear maps. In particular, we show:

(1) The sparsity achieved by [Kane-Nelson, SODA 2012] in the sparse Johnson-Lindenstrauss lemma is optimal up to a log(1/eps) factor.

(2) RIP_2 matrices preserving k-space vectors in R^n with the optimal number of rows must be dense as long as k < n / polylog(n).

## Faster Numerical Linear Algebra Algorithms Via Sparser Subspace Embeddings

## On Bilinear Complexity

For a set of polynomials F, we define their bilinear complexity as the smallest k so that F lies in an ideal generated by k bilinear polynomials. The main open problem is to estimate the bilinear complexity of the single polynomial $\sum_{i,j}x_i^2 y_j^2$. This question is related to the classical sum-of-squares problem as well as to problems in arithmetic circuit complexity. We will focus on related sets of polynomials and prove some lower and upper bounds on their bilinear complexity.

## The SOS (aka Lassere/Positivestellensatz/Sum-of-Squares) System Series

We will give an overview of this system, which has been at the center of recent algorithmic and proof complexity developments. We will give the definitions of the system (as a proof system for polynomial inequalities, and as an SDP-based algorithm), and basic upper and lower bounds for it. In particular we'll explain the recent SOS-proof of the hypercontractive inequality for the noisy hypercube of Barak et al., as well as the degree lower bounds for proving Tseitin and Knapsack tautologies of Grigoriev.

## Combinatorial PCPs with Short Proofs

The PCP theorem (Arora et. al., J. ACM 45(1,3)) asserts the existence of proofs that can be verified by reading a very small part of the proof. Since the discovery of the theorem, there has been a considerable work on improving the theorem in terms of the length of the proofs, culminating in the construction of PCPs of quasi-linear length, by Ben-Sasson and Sudan (SICOMP 38(2)) and Dinur (J. ACM 54(3)).

## Matching: A New Proof for an Ancient Algorithm

For all practical purposes, the Micali-Vazirani algorithm, discovered in 1980, is still the most efficient known maximum matching algorithm (for very dense graphs, slight asymptotic improvement can be obtained using fast matrix multiplication). However, this has remained a ``black box" result for the last 32 years. We hope to change this with the help of a recent paper giving a simpler proof and exposition of the algorithm:

http://arxiv.org/abs/1210.4594

## Delegation for Bounded Space

## Information Complexity and Exact Communication Bounds

In this talk we will discuss information complexity -- a measure of the amount of information Alice and Bob need to exchange to solve a problem over distributed inputs. We will present an information-theoretically optimal protocol for computing the AND of two bits distributed between Alice and Bob. We prove that the information complexity of AND is ~1.4923 bits. We use the optimal protocol and its properties to obtain tight bounds for the Disjointness problem, showing that the randomized communication complexity of Disjointness on n bits is ~0.4827n ± o(n).