The interval order polytope of a digraph

  • Rudolf Müller
  • Andreas S. Schulz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 920)


We introduce the interval order polytope of a digraph D as the convex hull of interval order inducing arc subsets of D. Two general schemes for producing valid inequalities are presented. These schemes have been used implicitly for several polytopes and they are applied here to the interval order polytope. It is shown that almost all known classes of valid inequalities of the linear ordering polytope can be explained by the two classes derived from these schemes. We provide two applications of the interval order polytope to combinatorial optimization problems for which to our knowledge no polyhedral descriptions have been given so far. One of them is related to analyzing DNA subsequences.


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  1. [Bal89]
    E. Balas. The asymmetric assignment problem and some new facets of the traveling salesman polytope on a directed graph. SIAM Journal on Discrete Mathematics, 2(4):425–451, 1989.Google Scholar
  2. [CF95]
    A. Caprara and M. Fischetti. Odd cut-sets, odd cycles and 0–1/2 Chvátal-Gomory cuts. Technical Report, DEIS, University of Bologna, Bologna, Italy, 1993, revised 1995.Google Scholar
  3. [Chv75]
    V. Chvátal. On certain polytopes associated with graphs. Journal of Combinatorial Theory Ser. B, 13:138–154, 1975.Google Scholar
  4. [CR93]
    S. Chopra and M. R. Rao. The partition problem. Mathematical Programming, 59:87–115, 1993.Google Scholar
  5. [Fis85]
    P. C. Fishburn. Interval Orders and Interval Graphs: A Study of Partially Ordered Sets. John Wiley & Sons, New York, 1985.Google Scholar
  6. [Gil90]
    I. Gilboa. A necessary but insufficient condition for the stochastic binary choice problem. Journal of Mathematical Psychology, 34:371–392, 1990.Google Scholar
  7. [GJ79]
    M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, 1979.Google Scholar
  8. [GJR85]
    M. Grötschel, M. Jünger, and G. Reinelt. Facets of the linear ordering polytope. Mathematical Programming, 33:43–60, 1985.Google Scholar
  9. [GKS92]
    M. C. Golumbic, H. Kaplan, and R. Shamir. Graph sandwich problems. Technical Report 270/92, Computer Science Dept., Tel Aviv University, Israel, 1992.Google Scholar
  10. [GKS94]
    M. C. Golumbic, H. Kaplan, and R. Shamir. On the complexity of DNA physical mapping. Advances in Applied Mathematics, 15:251–261, 1994.Google Scholar
  11. [GS93]
    M. C. Golumbic and R. Shamir. Complexity and algorithms for reasoning about time: A graph-theoretic approach. Journal of the Association for Computing Machinery, 40:1108–1133, 1993.Google Scholar
  12. [GW90]
    M. Grötschel and Y. Wakabayashi. Facets of the clique partitioning polytope. Mathematical Programming, 47(3):367–388, 1990.Google Scholar
  13. [LL94]
    J. Leung and J. Lee. More facets from fences for linear ordering and acyclic subgraph polytopes. Discrete Applied Mathematics, 50:185–200, 1994.Google Scholar
  14. [Möh90]
    R. H. Möhring. Graph problems related to gate matrix layout and PLA folding. In G. Tinhofer, E. Mayr, H. Noltemeier, and M. Syslo, editors, Computational Graph Theory, pages 17–52. Springer, 1990.Google Scholar
  15. [Möh94]
    R. H. Möhring, 1994. Personal communication.Google Scholar
  16. [MR89]
    R. H. Möhring and F. J. Radermacher. The order theoretic approach to scheduling: The deterministic case. In R. Slowinski and J. Weglarz, editors, Advances in Project Scheduling, pages 26–66. Elseviers Science Publication, 1989.Google Scholar
  17. [Mül93a]
    R. Müller. Bounds for linear VLSI problems. PhD thesis, Fachbereich Mathematik, Technische Universität Berlin, Berlin, Germany, 1993.Google Scholar
  18. [Mül93b]
    R. Müller. On the transitive acyclic subdigraph polytope. In G. Rinaldi and L. Wolsey, editors, Proceedings of the Third Conference on Integer Programming and Combinatorial Optimization, pages 463–477, 1993.Google Scholar
  19. [NW88]
    G. L. Nemhauser and L. A. Wolsey. Integer and Combinatorial Optimization. John Wiley & Sons, New York, 1988.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Rudolf Müller
    • 1
  • Andreas S. Schulz
    • 2
  1. 1.Institut für WirtschaftsinformatikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Fachbereich Mathematik (MA 6-1)Technische Universität BerlinBerlinGermany

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