Fooling Polytopes

TL;DR

This paper introduces a pseudorandom generator that effectively fools polytopes with many facets over binary spaces, significantly improving seed length efficiency and enabling deterministic approximation algorithms for integer programming problems.

Contribution

It presents a new pseudorandom generator with polylogarithmic seed length dependence on the number of facets, advancing the state of the art in polytope approximation.

Findings

Seed length is polylogarithmic in the number of facets

Enables deterministic quasipolynomial time algorithms for integer programming

Improves upon previous superlinear seed length bounds

Abstract

We give a pseudorandom generator that fools $m$-facet polytopes over $\{0,1\}^n$ with seed length $\mathrm{polylog}(m) \cdot \log n$. The previous best seed length had superlinear dependence on $m$. An immediate consequence is a deterministic quasipolynomial time algorithm for approximating the number of solutions to any $\{0,1\}$-integer program.