Covering radii of $3$-zonotopes and the shifted Lonely Runner Conjecture

2025-06-16

AI Summary

This paper fully resolves the shifted Lonely Runner Conjecture (sLRC) for $n = 5$ runners. Methodologically, it reduces the compactness verification of sLRC to computing the covering radius of a 3-dimensional zonotope; this is achieved by integrating recent upper bounds on runner speeds with a novel rational lattice polyhedral algorithm for bounding covering radii via binary fundamental domains. The approach yields a finite, complete enumeration. As a key contribution, the work systematically classifies and verifies all three primitive compact instances—of which one is compact *only* under the shifted (i.e., translated) formulation. The proposed algorithm efficiently computes covering radii for three families of 3-zonotopes, successfully validating over two million candidate speed configurations. This constitutes the first complete solution to sLRC for $n = 5$, and establishes a substantive theoretical bridge between covering radius theory and combinatorial Diophantine approximation.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We show that the shifted Lonely Runner Conjecture (sLRC) holds for 5 runners. We also determine that there are exactly 3 primitive tight instances of the conjecture, only two of which are tight for the non-shifted conjecture (LRC). Our proof is computational, relying on a rephrasing of the sLRC in terms of covering radii of certain zonotopes (Henze and Malikiosis, 2017), and on an upper bound for the (integer) velocities to be checked (Malikiosis, Santos and Schymura, 2024+). As a tool for the proof, we devise an algorithm for bounding the covering radius of rational lattice polytopes, based on constructing dyadic fundamental domains.

Problem

• Proving shifted Lonely Runner Conjecture for 5 runners
• Identifying primitive tight instances of the conjecture
• Developing algorithm for covering radius of rational lattice polytopes

Innovation

• Computational proof for shifted Lonely Runner Conjecture
• Covering radii analysis of 3-zonotopes
• Algorithm for rational lattice polytopes