On skeletons, diameters and volumes of metric polyhedra

  • Antoine Deza
  • Michel Deza
  • Komei Fukuda
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1120)


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.


Complete Graph Adjacency Relation Edge Connectivity Geometric Diameter Conic Hull 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Assouad P. and Deza M.: Metric subspaces of L 1. Publications mathématiques d'Orsay 3 (1982)Google Scholar
  2. 2.
    Avis D.: On the extreme rays of the metric cone. Canadian Journal of Mathematics XXXII 1 (1980) 126–144Google Scholar
  3. 3.
    Avis D.: In H. Imai ed. RIMS Kokyuroku A C Implementation of the Reverse Search Vertex Enumeration Algorithm. 872 (1994)Google Scholar
  4. 4.
    Avis D., Bremmer D. and Seidel R.: How good are convex hull algorithms. Computational Geometry: Theory and Applications (to appear)Google Scholar
  5. 5.
    Avis D. and Deza A.: Solitaire Cones. (in preparation)Google Scholar
  6. 6.
    Balinski M.: On the graph structure of convex polyhedra in n-space. Pacific Journal of Mathematics 11 (1961) 431–434Google Scholar
  7. 7.
    Barahona F. and Mahjoub R.: On the cut polytope. Mathematical Programming 36 (1986) 157–173Google Scholar
  8. 8.
    Bayer M. and Lee C.: Combinatorial aspects of convex polytopes. In P. Gruber and J. Wills eds. Handbook on Convex Geometry North Holland (1994) 485–534Google Scholar
  9. 9.
    Brouwer A., Cohen A. and Neumaier A.: Distance-Regular Graphs. Springer-Verlag, Berlin (1989)Google Scholar
  10. 10.
    Christof T. and Reinelt G.: Combinatorial optimization and small polytopes. To appear in Spanish Statistical and Operations Research Society 3 (1996)Google Scholar
  11. 11.
    Christof T. and Reinelt G.: Computing linear descriptions of combinatorial polytopes. (in preparation)Google Scholar
  12. 12.
    Deza A.: Metric polyhedra combinatorial structure and optimization. (in preparation)Google Scholar
  13. 13.
    Deza A. and Deza M.: The ridge graph of the metric polytope and some relatives. In T. Bisztriczky, P. McMullen, R. Schneider and A. Ivic Weiss eds. Polytopes: Abstract, Convex and Computational (1994) 359–372Google Scholar
  14. 14.
    Deza A. and Deza M.: The combinatorial structure of small cut and metric polytopes. In T. H. Ku ed. Combinatorics and Graph Theory, World Scientific Singapore (1995) 70–88Google Scholar
  15. 15.
    Deza M., Grishukhin V. and Laurent M.: The symmetries of the cut polytope and of some relatives. In P. Gritzmann and P Sturmfels eds. Applied Geometry and Discrete Mathematics, the ”Victor Klee Festschrift” DIMACS Series in Discrete Mathematics and Theoretical Computer Science 4 (1991) 205–220Google Scholar
  16. 16.
    Deza M. and Laurent M.: Facets for the cut cone I. Mathematical Programming 56 (2) (1992) 121–160CrossRefGoogle Scholar
  17. 17.
    Deza M. and Laurent M.: Applications of cut polyhedra. Journal of Computational and Applied Mathematics 55 (1994) 121–160 and 217–247Google Scholar
  18. 18.
    Deza M. and Laurent M.: New results on facets of the cut cone. R.C. Bose memorial issue of Journal of Combinatorics, Information and System Sciences 17 (1–2) (1992) 19–38Google Scholar
  19. 19.
    Deza M., Laurent M. and Poljak S.: The cut cone III: on the role of triangle facets. Graphs and Combinatorics 8 (1992) 125–142Google Scholar
  20. 20.
    Fukuda K.: cdd reference manual, version 0.56. ETH Zentrum, Zürich, Switzerland (1995)Google Scholar
  21. 21.
    Grishukhin V. P.: Computing extreme rays of the metric cone for seven points. European Journal of Combinatorics 13 (1992) 153–165Google Scholar
  22. 22.
    Iri M.: On an extension of maximum-flow minimum-cut theorem to multicommodity flows. Journal of the Operational Society of Japan 13 (1970–1971) 129–135Google Scholar
  23. 23.
    Laurent M.: Graphic vertices of the metric polytope. Discrete Mathematics 145 (1995) (to appear)Google Scholar
  24. 24.
    Laurent M. and Poljak S.: The metric polytope. In E. Balas, G. Cornuejols and R. Kannan eds. Integer Programming and Combinatorial Optimization Carnegie Mellon University, GSIA, Pittsburgh (1992) 274–285Google Scholar
  25. 25.
    Murty K. G. and Chung S. J.: Segments in enumerating faces. Mathematical Programming 70 (1995) 27–45Google Scholar
  26. 26.
    Onaga K. and Kakusho O.: On feasibility conditions of multicommodity flows in networks. IEEE Trans. Circuit Theory 18 (1971) 425–429Google Scholar
  27. 27.
    Padberg M.: The boolean quadric polytope: some characteristics, facets and relatives. Mathematical Programming 45 (1989) 139–172CrossRefGoogle Scholar
  28. 28.
    Plesník J.: Critical graphs of given diameter. Acta Math. Univ. Comenian 30 (1975) 71–93Google Scholar
  29. 29.
    Poljak S. and Tuza Z.: Maximum Cuts and Large Bipartite Subgraphs. In W. Cook, L. Lovasz and P. D. Seymour eds. DIMACS 20 (1995) 181–244Google Scholar
  30. 30.
    Trubin V.: On a method of solution of integer linear problems of a special kind. Soviet Mathematics Doklady 10 (1969) 1544–1546Google Scholar
  31. 31.
    Ziegler G. M.: Lectures on Polytopes. Graduate Texts in Mathematics 152 Springer-Verlag, New York, Berlin, Heidelberg (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 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

Personalised recommendations