Computing the covering radius of a polytope with an application to lonely runners

TL;DR

This paper introduces a new, efficient algorithm for computing the covering radius of rational polytopes, with applications to the Lonely Runner Conjecture, successfully solving a previously open case.

Contribution

It presents a simpler, more efficient algorithm for the covering radius problem and applies it to a geometric variant of the Lonely Runner Conjecture.

Findings

New algorithm for covering radius computation

First open case of three runners with individual starting points solved

Algorithm is simpler and more efficient than previous methods

Abstract

We are concerned with the computational problem of determining the covering radius of a rational polytope. This parameter is defined as the minimal dilation factor that is needed for the lattice translates of the correspondingly dilated polytope to cover the whole space. As our main result, we describe a new algorithm for this problem, which is simpler, more efficient and easier to implement than the only prior algorithm of Kannan (1992). Motivated by a variant of the famous Lonely Runner Conjecture, we use its geometric interpretation in terms of covering radii of zonotopes, and apply our algorithm to prove the first open case of three runners with individual starting points.