On skeletons, diameters and volumes of metric polyhedra

  • Antoine Deza
  • Michel Deza
  • Komei Fukuda
Conference paper

DOI: 10.1007/3-540-61576-8_78

Part of the Lecture Notes in Computer Science book series (LNCS, volume 1120)
Cite this paper as:
Deza A., Deza M., Fukuda K. (1996) On skeletons, diameters and volumes of metric polyhedra. In: Deza M., Euler R., Manoussakis I. (eds) Combinatorics and Computer Science. Lecture Notes in Computer Science, vol 1120. Springer, Berlin, Heidelberg

Abstract

We survey and present new geometric and combinatorial properties, of some polyhedra with application in combinatorial optimization, for example, the max-cut and multicommodity flow problems. Namely we consider the volume, symmetry group, facets, vertices, face lattice, diameter, adjacency and incidence relations and connectivity of the metric polytope and its relatives. In particular, using its large symmetry group, we completely describe all the 13 orbits which form the 275 840 vertices of the 21-dimensional metric polytope on 7 nodes and their incidence and adjacency relations. The edge connectivity, the i-skeletons and a lifting procedure valid for a large class of vertices of the metric polytope are also given. Finally, we present an ordering of the facets of a polytope, based on their adjacency relations, for the enumeration of its vertices by the double description method.

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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Antoine Deza
    • 1
  • Michel Deza
    • 2
  • Komei Fukuda
    • 3
    • 4
  1. 1.Department of Mathematical and Computing SciencesTokyo Institute of TechnologyTokyoJapan
  2. 2.Ecole Normale Supérieure, Département de Mathématiques et d'InformatiqueCNRSParisFrance
  3. 3.Institute for Operations ResearchETH ZürichZürichSwitzerland
  4. 4.Graduate School of Systems ManagementUniversity of TsukubaTokyoJapan

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