A PRG for Lipschitz Functions of Polynomials with Applications to Sparsest Cut

TL;DR

This paper introduces improved pseudorandom generators for Lipschitz functions of low-degree polynomials, enabling better approximation algorithms for the sparsest cut problem by constructing instances with larger integrality gaps.

Contribution

It provides the first nontrivial PRG for Lipschitz functions of degree O(log n) polynomials with near-optimal seed length, and applies this to improve lower bounds for sparsest cut.

Findings

New PRG with seed length (log n)~O(d^2/eps^2) for Lipschitz functions

Constructed sparsest cut instances with exponential integrality gaps

Enhanced understanding of SDP performance in graph partitioning

Abstract

We give improved pseudorandom generators (PRGs) for Lipschitz functions of low-degree polynomials over the hypercube. These are functions of the form psi(P(x)), where P is a low-degree polynomial and psi is a function with small Lipschitz constant. PRGs for smooth functions of low-degree polynomials have received a lot of attention recently and play an important role in constructing PRGs for the natural class of polynomial threshold functions. In spite of the recent progress, no nontrivial PRGs were known for fooling Lipschitz functions of degree O(log n) polynomials even for constant error rate. In this work, we give the first such generator obtaining a seed-length of (log n)\tilde{O}(d^2/eps^2) for fooling degree d polynomials with error eps. Previous generators had an exponential dependence on the degree. We use our PRG to get better integrality gap instances for sparsest cut, a fundamental problem in graph theory with many applications in graph optimization. We give an instance of uniform sparsest cut for which a powerful semi-definite relaxation (SDP) first introduced by Goemans and Linial and studied in the seminal work of Arora, Rao and Vazirani has an integrality gap of exp(\Omega((log log n)^{1/2})). Understanding the performance of the Goemans-Linial SDP for uniform sparsest cut is an important open problem in approximation algorithms and metric embeddings and our work gives a near-exponential improvement over previous lower bounds which achieved a gap of \Omega(log log n).