Computational Complexity 7 JAN, 2012

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

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

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).

💡 Research Summary

The paper tackles two intertwined problems: constructing a pseudorandom generator (PRG) that fools Lipschitz functions of low‑degree polynomials, and leveraging this PRG to obtain stronger integrality‑gap lower bounds for the uniform sparsest‑cut problem under the Goemans‑Linial semidefinite programming (SDP) relaxation.

Background.
A Lipschitz function of a polynomial has the form ψ(P(x)), where P : {−1,1}ⁿ → ℝ is a degree‑d polynomial and ψ : ℝ → ℝ has Lipschitz constant L. Such functions appear in learning theory (e.g., smoothed polynomial threshold functions), approximation algorithms, and metric embeddings. Prior PRGs for smooth functions of low‑degree polynomials (e.g., those based on bounded independence and the invariance principle) achieve seed lengths that grow exponentially in d, making them ineffective when d = Θ(log n). Consequently, no non‑trivial PRG was known for Lipschitz functions of degree‑O(log n) polynomials, even for constant error ε.

Main technical contribution – a block‑wise PRG.
The authors introduce a “divide‑and‑conquer” construction. The n input bits are partitioned into t blocks B₁,…,B_t of equal size n/t. For each block they instantiate a PRG that guarantees (d′, ε′)-wise independence, where d′ = d/t. The blocks are generated independently, using disjoint seed portions. The key observations are:

  1. Degree reduction. By restricting attention to a block, the effective degree of the polynomial restricted to that block drops from d to d′ = d/t. This allows the use of existing PRGs whose seed length scales as O((d′)²·log(1/ε′)).

  2. Error accumulation. Since ψ is L‑Lipschitz, the error contributed by a single block is at most O(L·ε′). By a union bound over the t blocks, the total error is O(t·L·ε′). Setting ε′ = ε/(t·L) guarantees overall error ≤ ε.

  3. Seed length analysis. The seed for each block costs O((d/t)²·log(t·L/ε)). Adding the t·log n bits needed to specify the block‑wise independent seeds yields a total seed length
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