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Edges versus circuits: a hierarchy of diameters in polyhedra

  • S. Borgwardt , J. A. De Loera EMAIL logo and E. Finhold
Published/Copyright: October 13, 2016
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Advances in Geometry
From the journal Advances in Geometry Volume 16 Issue 4

Abstract

The study of the graph diameter of polytopes is a classical open problem in polyhedral geometry and the theory of linear optimization. In this paper we continue the investigation initiated in [5] by introducing a vast hierarchy of generalizations to the notion of graph diameter. This hierarchy provides some interesting lower bounds for the usual graph diameter. After explaining the structure of the hierarchy and discussing these bounds, we focus on clearly explaining the differences and similarities among the many diameter notions of our hierarchy. Finally, we fully characterize the hierarchy in dimension two. It collapses into fewer categories, for which we exhibit the ranges of values that can be realized as diameters.

Keywords: Diameter of polyhedra; graph of polyhedra; one-skeleton of polytopes; matroids
MSC 2010: 52B05; 52B55; 52B40; 52C40; 52C45; 90C05; 90C49

Communicated by: M. Joswig

Acknowledgements

We are grateful for the comments we receive from Raymond Hemmecke, Jon Lee, and Edward Kim regarding some of these constructions. We are also grateful to Jake Miller for his help during the crafting of this paper.

Funding

The first author was supported by the Humboldt Foundation. The second author was supported by an NSA grant. The third author acknowledges the support from the graduate program TopMath of the Elite Network of Bavaria and the TopMath Graduate Center of TUM Graduate School at Technische Universität München.

References

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Received: 2014-11-18
Revised: 2016-3-27
Published Online: 2016-10-13
Published in Print: 2016-10-1

© 2016 by Walter de Gruyter Berlin/Boston

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https://doi.org/10.1515/advgeom-2016-0020
Keywords for this article
Diameter of polyhedra; graph of polyhedra; one-skeleton of polytopes; matroids

Articles in the same Issue

  1. Research Article
  2. Comparison theorems on smooth metric measure spaces with boundary
  3. Research Article
  4. Hilbert schemes of some threefold scrolls over 𝔽e
  5. Research Article
  6. On caustics by reflection of algebraic surfaces
  7. Research Article
  8. Optimal cover of a disk with three smaller congruent disks
  9. Research Article
  10. Balanced and absorbing subsets with empty interior
  11. Research Article
  12. Hyperovals arising from a Singer group action on H(3,q2), q even
  13. Research Article
  14. A fixed point theorem in Euclidean buildings
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  16. Positive polynomials on nondegenerate basic semi-algebraic sets
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  20. Monads for framed sheaves on Hirzebruch surfaces, Adv. Geom. 15 (2015), 55–76
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