Advertisement

Mathematical Programming

, Volume 33, Issue 1, pp 43–60 | Cite as

Facets of the linear ordering polytope

  • Martin Grötschel
  • Michael Jünger
  • Gerhard Reinelt
Article

Abstract

LetD n be the complete digraph onn nodes, and letP LO n denote the convex hull of all incidence vectors of arc sets of linear orderings of the nodes ofD n (i.e. these are exactly the acyclic tournaments ofD n ). We show that various classes of inequalities define facets ofP LO n , e.g. the 3-dicycle inequalities, the simplek-fence inequalities and various Möbius ladder inequalities, and we discuss the use of these inequalities in cutting plane approaches to the triangulation problem of input-output matrices, i.e. the solution of permutation resp. linear ordering problems.

Key words

Facets of Polyhedra Linear Ordering Problem Triangulation Problem Permutation Problem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. V.J. Bowman, “Permutation polyhedra”, SIAMJournal on Applied Mathematics 22 (1972) 580–589.Google Scholar
  2. M.R. Garey and D.S. Johnson,Computers and intractability: A guide to the theory of NP-completeness (Freeman, San Francisco, 1979).Google Scholar
  3. M. Grötschel, M. Jünger and G. Reinelt, “On the acyclic subgraph polytope”, this volume, pp. 28–42.Google Scholar
  4. M. Grötschel, M. Jünger and G. Reinelt, “A cutting plane algorithm for the linear ordering problem”,Operations Research 32 (1984) 1195–1220.Google Scholar
  5. M. Grötschel, M. Jünger and G. Reinelt, “Optimal triangulation of large real-world input-outputmatrices”,Statistische Hefte 25 (1984) 261–295.Google Scholar
  6. B. Korte and W. Oberhofer, “Zwei Algorithmen zur Lösung eines komplexen Reinhenfolgeproblems”,Unternehmensforschung 12 (1968) 217–362.Google Scholar
  7. B. Korte and W. Oberhofer, “Zur Triangulation von Input-Output Matrizen”,Jahrbücher für Nationalökonomie und Statistik 182 (1969) 398–433.Google Scholar
  8. H.W. Lenstra, Jr., “The acyclic subgraph problem”, Report BW26, Mathematisch Centrum (Amsterdam, 1973).Google Scholar
  9. J.F. Mascotorchino and P. Michaud,Optimisation en analyse ordinale des données (Masson, Paris, 1979).Google Scholar
  10. H. Wessels, “Triangulation und Blocktriangulation von Input-Output Tabellen”,Deutsches Institut für Wirtschaftsforschung: Beiträge aur Strukturforschung, Heft 63, Berlin, 1981.Google Scholar
  11. H.P. Young, “On permutations and permutation polytopes”,Mathematical Programming Study 8 (1978) 128–140.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1985

Authors and Affiliations

  • Martin Grötschel
    • 1
  • Michael Jünger
    • 1
  • Gerhard Reinelt
    • 1
  1. 1.Institut für MathematikUniversität AugsburgAugsburgFR Germany

Personalised recommendations