On the Skeleton of the Metric Polytope

  • Antoine Deza
  • Komei Fukuda
  • Dmitrii Pasechnik
  • Masanori Sato
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2098)

Abstract

We consider convex polyhedra with applications to well-known combinatorial optimization problems: the metric polytope m n and its relatives. For n ≤ 6 the description of the metric polytope is easy as m n has at most 544 vertices partitioned into 3 orbits; m 7 - the largest previously known instance - has 275 840 vertices but only 13 orbits. Using its large symmetry group, we enumerate orbitwise 1 550 825 600 vertices of the 28-dimensional metric polytope m s . The description consists of 533 orbits and is conjectured to be complete. The orbitwise incidence and adjacency relations are also given. The skeleton of m s could be large enough to reveal some general features of the metric polytope on n nodes. While the extreme connectivity of the cuts appears to be one of the main features of the skeleton of m n , we conjecture that the cut vertices do not form a cut-set. The combinatorial and computational applications of this conjecture are studied. In particular, a heuristic skipping the highest degeneracy is presented.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Antoine Deza
    • 1
    • 5
  • Komei Fukuda
    • 2
  • Dmitrii Pasechnik
    • 3
  • Masanori Sato
    • 4
  1. 1.Prediction and ControlInstitute of Statistical MathematicsTokyoJapan
  2. 2.Institute for Operations ResearchETH ZürichZürichSwitzerland
  3. 3.Operations Research, TWIDelft University of TechnologyNetherlands
  4. 4.Tokyo Institute of Technology, Math. and Comput. Sci.TokyoJapan
  5. 5.Centre d’Analyse et de Mathématique SocialesEHESSParisFrance

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