A Spectral Approach to Polytope Diameter

TL;DR

This paper establishes new upper bounds on the graph diameters of polytopes using spectral methods, applicable in worst-case and smoothed analysis scenarios, highlighting the role of spectral gaps and log-concavity.

Contribution

It introduces spectral techniques to bound polytope diameters, improving known results and providing probabilistic bounds under Gaussian perturbations.

Findings

Improved worst-case diameter bounds based on integer constraints.

High-probability existence of a giant component with polynomial diameter after Gaussian noise.

Spectral gaps derived from log-concavity are key to the bounds.

Abstract

We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint matrix, which in some cases improves previously known results. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high probability there exists a "giant component" of vertices, with measure $1-o(1)$ and polynomial diameter. Both bounds rely on spectral gaps -- of a certain Schr\"odinger operator in the first case, and a certain continuous time Markov chain in the second -- which arise from the log-concavity of the volume of a simple polytope in terms of its slack variables.