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.
Research supported by Japanese Ministry of Education, Science and Culture for the first author.
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References
Assouad P. and Deza M.: Metric subspaces of L 1. Publications mathématiques d'Orsay 3 (1982)
Avis D.: On the extreme rays of the metric cone. Canadian Journal of Mathematics XXXII 1 (1980) 126–144
Avis D.: In H. Imai ed. RIMS Kokyuroku A C Implementation of the Reverse Search Vertex Enumeration Algorithm. 872 (1994)
Avis D., Bremmer D. and Seidel R.: How good are convex hull algorithms. Computational Geometry: Theory and Applications (to appear)
Avis D. and Deza A.: Solitaire Cones. (in preparation)
Balinski M.: On the graph structure of convex polyhedra in n-space. Pacific Journal of Mathematics 11 (1961) 431–434
Barahona F. and Mahjoub R.: On the cut polytope. Mathematical Programming 36 (1986) 157–173
Bayer M. and Lee C.: Combinatorial aspects of convex polytopes. In P. Gruber and J. Wills eds. Handbook on Convex Geometry North Holland (1994) 485–534
Brouwer A., Cohen A. and Neumaier A.: Distance-Regular Graphs. Springer-Verlag, Berlin (1989)
Christof T. and Reinelt G.: Combinatorial optimization and small polytopes. To appear in Spanish Statistical and Operations Research Society 3 (1996)
Christof T. and Reinelt G.: Computing linear descriptions of combinatorial polytopes. (in preparation)
Deza A.: Metric polyhedra combinatorial structure and optimization. (in preparation)
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) 359–372
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) 70–88
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) 205–220
Deza M. and Laurent M.: Facets for the cut cone I. Mathematical Programming 56 (2) (1992) 121–160
Deza M. and Laurent M.: Applications of cut polyhedra. Journal of Computational and Applied Mathematics 55 (1994) 121–160 and 217–247
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) 19–38
Deza M., Laurent M. and Poljak S.: The cut cone III: on the role of triangle facets. Graphs and Combinatorics 8 (1992) 125–142
Fukuda K.: cdd reference manual, version 0.56. ETH Zentrum, Zürich, Switzerland (1995)
Grishukhin V. P.: Computing extreme rays of the metric cone for seven points. European Journal of Combinatorics 13 (1992) 153–165
Iri M.: On an extension of maximum-flow minimum-cut theorem to multicommodity flows. Journal of the Operational Society of Japan 13 (1970–1971) 129–135
Laurent M.: Graphic vertices of the metric polytope. Discrete Mathematics 145 (1995) (to appear)
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) 274–285
Murty K. G. and Chung S. J.: Segments in enumerating faces. Mathematical Programming 70 (1995) 27–45
Onaga K. and Kakusho O.: On feasibility conditions of multicommodity flows in networks. IEEE Trans. Circuit Theory 18 (1971) 425–429
Padberg M.: The boolean quadric polytope: some characteristics, facets and relatives. Mathematical Programming 45 (1989) 139–172
Plesník J.: Critical graphs of given diameter. Acta Math. Univ. Comenian 30 (1975) 71–93
Poljak S. and Tuza Z.: Maximum Cuts and Large Bipartite Subgraphs. In W. Cook, L. Lovasz and P. D. Seymour eds. DIMACS 20 (1995) 181–244
Trubin V.: On a method of solution of integer linear problems of a special kind. Soviet Mathematics Doklady 10 (1969) 1544–1546
Ziegler G. M.: Lectures on Polytopes. Graduate Texts in Mathematics 152 Springer-Verlag, New York, Berlin, Heidelberg (1995)
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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. CCS 1995. Lecture Notes in Computer Science, vol 1120. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61576-8_78
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DOI: https://doi.org/10.1007/3-540-61576-8_78
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