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Computing the Covering Radius of a Polytope with an Application to Lonely Runners

Abstract

We study 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.

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Acknowledgments

We thank Jörg M. Wills and Gennadiy Averkov for thorough reading of an earlier version of the manuscript, and for providing valuable comments and suggestions. We thank the anonymous referees for very careful reading and for suggestions that improved the quality of the presentation of our material.

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Correspondence to Matthias Schymura.

Additional information

Jana Cslovjecsek and Matthias Schymura were supported by the Swiss National Science Foundation (SNSF) within the project Lattice Algorithms and Integer Programming (Nr. 185030). Márton Naszódi was supported by the National Research, Development and Innovation Fund (NRDI) grants K119670 and KKP-133864, the Bolyai Scholarship of the Hungarian Academy of Sciences and the New National Excellence Programme and the TKP2020-NKA-06 program provided by the NRDI.

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Cslovjecsek, J., Malikiosis, R.D., Naszódi, M. et al. Computing the Covering Radius of a Polytope with an Application to Lonely Runners. Combinatorica (2022). https://doi.org/10.1007/s00493-020-4633-8

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  • DOI: https://doi.org/10.1007/s00493-020-4633-8

Mathematics Subject Classification (2010)

  • 11H31
  • 52C17
  • 11J25
  • 52B55