Skip to main content
SpringerLink
Log in

Acyclic Subdigraphs and Linear Orderings: Polytopes, Facets, and a Cutting Plane Algorithm

  • Chapter
Graphs and Order

Part of the book series: NATO ASI Series ((ASIC,volume 147))

Abstract

We study the acyclic subdigraph problem and the linear ordering problem from a polyhedral point of view. Insights into the facet structure of polytopes associated with these problems lead to the formulation and implementation of a cutting plane algorithm for the linear ordering problem.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

VIII. Bibliography (relevant for the subject, but not necessarily mentioned in the paper)

  • A. Bachem & M. Grötschel (1982) “New Aspects of Polyhedral Theory”, in: B. Korte (ed.) “Modern Applied Mathematics”. North-Holland, Amsterdam, 1982, 51–106.

    Google Scholar 

  • K. Boenchendorf (1982) “Reihenfolgenprobleme/Mean-Flow-Time Sequencing” Mathematical Systems in Economics 74, Verlagsgruppe Athenaum/Hain/ Scriptor/Hanstein, 1982.

    Google Scholar 

  • V. J. Bowman (1972) “Permutation Polyhedra” SIAM J. Appl. Math., Vol. 22, No. 4 (1972), 580–589.

    Article  MathSciNet  MATH  Google Scholar 

  • J. S. de Cani (1969) “Maximum Likelyhood Paired Comparison Ranking by Linear Programming”. Biometrika 56(1969), 537–545.

    Article  MathSciNet  MATH  Google Scholar 

  • V. Chvátal (1973) “Edmonds Polytope and Weakly Hamiltonian Graphs”. Mathematical Programming 5 (1973), 29–40.

    Article  MathSciNet  MATH  Google Scholar 

  • H. Crowder, E. L. Johnson & M. W. Padberg (1981) “Solving Large Scale 0-1-Programming Problems”. Research Report RC8888 (38963), IBM T. J. Watson Research Center, Yorktown Heights, New York, 1981.

    Google Scholar 

  • H. Crowder & M. W. Padberg (1980) “Solving Large Scale Symmetric Travelling Salesman Problems”. Management Science 26 (1980), 495–509.

    Article  MathSciNet  MATH  Google Scholar 

  • G. B. Dantzig (1963) “Linear Programming and Extensions” Princeton University Press, Princeton, New Jersey, 1963.

    MATH  Google Scholar 

  • G. B. Dantzig, D. R. Fulkerson & S. M. Johnson (1954) “Solution of a Large-Scale Travelling Salesman Problem”. Operations Research 2 (1954), 393–410.

    Article  MathSciNet  Google Scholar 

  • R. W. Floyd (1962) “Algorithm 97: Shortest Path”. Comm. ACM, 5, No. 6 (1962), 345.

    Article  Google Scholar 

  • M. R. Garey & D. S. Johnson (1979) “Computers and Intractability: A Guide to the Theory of NP-Completeness”. Freeman, San Francisco, 1979.

    MATH  Google Scholar 

  • R. Giles & W. R. Pulleyblank (1979) “Total Dual Integrality and Integer Polyhedra”. Linear Algebra and its Applications 25 (1979), 191–196

    Article  MathSciNet  MATH  Google Scholar 

  • F. Glover, T. Klastorin & D. Klingman (1974) “Optimal Weighted Ancestry Relationships”. Management Science 20 (1974), 1190–1193.

    Article  MATH  Google Scholar 

  • M. Grötschel (1977) “Polyedrische Charakterisierungen kombinatorischer Optimierungsprobleme” Verlag Anton Hain, Meisenheim am Glan, 1977.

    MATH  Google Scholar 

  • M. Grötschel (1983) “Polyhedral Combinatorics: An Annotated Bibliography”. Preprint No. 15, Mathematisches Institut, Universität Augsburg, 1983.

    Google Scholar 

  • M. Grötschel, M. Jünger & G. Reinelt (1982a) “On the Acyclic Subgraph Polytope”. Report No. 82215-OR, Institut für Operations Research, Universität Bonn, 1982.

    Google Scholar 

  • M. Grötschel, M. Jünger & G. Reinelt (1982b) “Facets of the Linear Ordering Polytope”. Report No. 82217-OR, Institut für Operations Research, Universität Bonn, 1982.

    Google Scholar 

  • M. Grötschel, M. Jünger & G. Reinelt (1983a) “A Cutting Plane Algorithm for the Linear Ordering Problem. Preprint No. 8, Mathematisches Institut, Universität Augsburg, 1983, to appear in Operations Research.

    Google Scholar 

  • M. Grötschel, M. Jünger & G. Reinelt (1983b) “Optimal Triangulation of Large Real-World Input-Output Matrices. Preprint No. 9, Mathematisches Institut, Universität Augsburg, 1983, to appear in Statistische Jahreshefte.

    Google Scholar 

  • M. Grötschel, L. Lovász & A. Schrijver (1981) “The Ellipsoid Method and its Consequences in Combinatorial Optimization”. Combinatorica 1 (1981), 169–197.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Grötschel & M. W. Padberg (1979a) “On the Symmetric Travelling Salesman Problem I: Inequalities”. Mathematical Programming 16 (1979), 265–280.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Grötschel & M. W. Padberg (1979b) “On the Symmetric Travelling Salesman Problem II: Lifting Theorems and Facets”. Mathematical Programming 16 (1979), 281–302.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Grötschel & M. W. Padberg (1983a) “Polyhedral Aspects of the Travelling Salesman Problem I: Theory” in: E. L. Lawler, J. K. Lenstra & A. H. G. Rinnooy Kan (eds.) “The Travelling Salesman Problem” John Wiley and Sons, London, New York, 1983.

    Google Scholar 

  • M. Grötschel & M. W. Padberg (1983b) “Polyhedral Aspects of the Travelling Salesman Problem II: Computation” in: E. L. Lawler, J. K. Lenstra & A. H. G. Rinnooy Kan (eds.) “The Travelling Salesman Problem”. John Wiley and Sons, London, New York, 1983.

    Google Scholar 

  • M. Grötschel & W. R. Pulleyblank (1981) “Clique Tree Inequalities and the Symmetric Travelling Salesman Problem”. Report No. 81196-OR, Institut für Operations Research, Universität Bonn, 1981.

    Google Scholar 

  • P. L. Hammer, E. L. Johnson & U. N. Peled (1975) “Facets of Regular 0-1-Polytopes” Mathematical Programming 8 (1975), 179–206.

    Article  MathSciNet  MATH  Google Scholar 

  • K. Helbig Hansen & J. Krarup (1974) “Improvements of the Held-Karp Algorithm for the Symmetric Travelling Salesman Problem”. Mathematical Programming 7 (1974), 87–96.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Held & R. M. Karp (1970) “The Travelling Salesman Problem and Minimum Spanning Trees”. Operations Research 18 (1970), 1138–1182.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Held & R. M. Karp (1971) “The Travelling Salesman Problem and Minimum Spanning Trees: Part II”. Mathematical Programming 1 (1971), 6–26.

    Article  MathSciNet  MATH  Google Scholar 

  • A. J. Hoffman & J. B. Kruskal (1956) “Integral Boundary Points of Convex Polyhedra”. in: H. W. Kuhn & A. W. Tucker (eds.) “Linear Inequalities and Related Systems” Princeton, University Press, Princeton, New Jersey (1956), 223–246.

    Google Scholar 

  • E. L. Johnson & U. Suhl (1980) “Experiments in Integral Programming”. Discrete Applied Mathematics 2 (1980), 39–55.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Jünger (1984) “Polyhedral Combinatorics and the Acyclic Subdigraph Problem”, Mathematisches Institut, Universität Augsburg, to appear.

    Google Scholar 

  • R. Kaas (1981) “A Branch and Bound Algorithm for the Acyclic Subgraph Problem”. European Journal of Operational Research 8 (1981), 355–362.

    Article  MathSciNet  MATH  Google Scholar 

  • R. M. Karp & C. H. Papadimitriou (1980) “On Linear Characterizations of Combinatorial Optimization Problems”. Proc. of the 21st Annual Symp. on the Foundations of Computer Science, IEEE (1980), 1–9.

    Google Scholar 

  • L. G. Khachian (1979) “A Polynomial Algorithm in Linear Programming”. Doklady Academii Nauk SSSR 244 (1979) (English translation: Soviet Math. Dokl. 20 (1979), 191–194).

    Google Scholar 

  • B. Korte & W. Oberhofer (1968) “Zwei Algorithmen zur Lösung eines komplexen Reihenfolgeproblems”. Unternehmensforschung 12 (1969), 217–231.

    Article  MATH  Google Scholar 

  • B. Korte & W. Oberhofer (1969) “Zur Triangulation von Input-Output Matrizen”. Jb. f. Nat. Ök. u. Stat. 182 (1969), 398–433.

    Google Scholar 

  • W. Krelle (1964) in: F. Neumark (ed.) “Strukturwandlungen einer wachsenden Wirtschaft”. Schriften des Vereins für Socialpolitik, NF Band 30/11 (1964), 1059–1063.

    Google Scholar 

  • K. Kuratowski (1930) “Sur le Probleme des Courbes Gauches en Topologie” Fund. Math. 15 (1930), 271–283.

    MATH  Google Scholar 

  • A. H. Land & A. G. Doig (1960) “An Automatic Method for Solving Discrete Programming Problems” Econometrica 28 (1960), 497–520.

    Article  MathSciNet  MATH  Google Scholar 

  • E. L. Lawler (1976) “Combinatorial Optimization: Networks and Matroids”. Holt, Rinehart and Winston, 1976.

    MATH  Google Scholar 

  • E. L. Lawler, J. K. Lenstra & A. H. G. Rinnooy Kan (eds.) (1983) “The Travelling Salesman Problem” John Wiley and Sons, London, New York, 1983.

    Google Scholar 

  • H. W. Lenstra Jr. (1973) “The Acyclic Subgraph Problem”. Report BW26, Mathematisch Centrum, Amsterdam, 1973.

    Google Scholar 

  • J. K. Lenstra (1977) “Sequencing by Enumerative Methods”. Mathematical Centre Tracts 69, Mathematisch Centrum, Amsterdam, 1977.

    Google Scholar 

  • L. Lovász (1976) “On Two Minimax Theorems in Graph Theory”. J. Combinatorial Theory (B) 21 (1976), 96–103.

    Article  MATH  Google Scholar 

  • C. L. Lucchesi (1976) “A Minimax Equality for Directed Graphs”. Doctoral Thesis, University of Waterloo, Waterloo, Ontario, 1976.

    Google Scholar 

  • C. L. Lucchesi & D. H. Younger (1978) “A Minimax Relation for Directed Graphs” J. London Math. Soc. (2) 17 (1978), 369–374.

    Article  MathSciNet  MATH  Google Scholar 

  • J. F. Marcotorchino & P. Michaud (1979) “Optimisation en Analyse Ordinale des Données” Masson, Paris, 1979.

    Google Scholar 

  • M. W. Padberg (1979) “Covering, Packing and Knapsack Problems”. Annals of Discrete Mathematics 4 (1979), 265–287.

    Article  MathSciNet  MATH  Google Scholar 

  • M. W. Padberg & S. Hong (1980) “On the Symmetric Travelling Salesman Problem: A Computational Study”. Mathematical Programming Studies 12 (1980), 78–107.

    MathSciNet  MATH  Google Scholar 

  • M. W. Padberg & M. R. Rao (1981) “The Russian Method for Linear Inequalities III: Bounded Integer Programming”. INRIA, Rapport de Recherche, Rocquencourt, revised May 1981, to appear in Mathematical Programming Studies.

    Google Scholar 

  • M. W. Padberg & M. R. Rao (1982) “Odd Minimum Cut Sets and b-Matchings”. Mathematics of Operations Research 7 (1982), 67–80.

    Article  MathSciNet  MATH  Google Scholar 

  • C. H. Papadimitriou & K. Steiglitz (1982) “Combinatorial Optimizations: Algorithms and Complexity” Prentice Hall, Englewood Cliffs, New Jersey, 1982.

    MATH  Google Scholar 

  • W. R. Pulleyblank (1973) “Facets of Matching Polyhedra” Doctoral Thesis, University of Waterloo, Waterloo, Ontario, 1973.

    Google Scholar 

  • W. R. Pulleyblank (1983) “Polyhedral Combinatorics”, in: A. Bachem, M. Grötschel & B. Korte (eds.) “Mathematical Programming, Bonn 1982—The State of the Art” Springer Verlag, Berlin (1983), 312–345.

    Google Scholar 

  • W. R. Pulleyblank & J. Edmonds (1974) “Facets of 1-Matching Polyhedra”. in: C. Berge & D. Ray-Chaudhuri (eds.) “Hypergraph Seminar”. Springer Verlag, Berlin (1974), 214–242.

    Chapter  Google Scholar 

  • G. Reinelt (1984) “The Linear Ordering Problem: Algorithms and Applications”. Mathematisches Institut, Universität Augsburg, to appear.

    Google Scholar 

  • A. Schrijver (1981) “On Total Dual Integrality”. Linear Algebra and its Applications 38 (1981), 27–32.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Schrijver (1983) “Min-Max Results in Combinatorial Optimization”, in: A. Bachem, M. Grötschel & B. Korte (eds.) “Mathematical Programming, Bonn 1982—The State of the Art”. Springer Verlag, Berlin (1983), 439–500.

    Google Scholar 

  • A. Schrijver (1984) “Polyhedral Combinatorics” John Wiley and Sons, New York, to appear.

    Google Scholar 

  • P. Slater (1961) “Inconsistencies in a Schedule of Paired Comparisons”. Biometrika 48 (1961), 303–312.

    Google Scholar 

  • S. Warshall (1962) “A Theorem on Boolean Matrices”. J. ACM, 9 No. 1 (1962), 11–12.

    Article  MathSciNet  MATH  Google Scholar 

  • H. Wessels (1981) “Triangulation und Blocktriangulation von Input-Output Tabellen. Deutsches Institut für Wirtschaftsforschung, Beiträge zur Strukturforschung, Heft 63, Berlin, 1981.

    Google Scholar 

  • H. Weyl (1935) “Elementare Theorie der konvexen Polyeder” Commentarii Mathematici Helvetici 7 (1935), 290–306.

    Article  MathSciNet  MATH  Google Scholar 

  • H. P. Young (1978) “On Permutations and Permutation Polytopes” Mathematical Programming Study 8 (1978), 128–140.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik Lehrstuhl für Angewandte Mathematik II, Universität Augsburg, Memminger Str. 6, 8900, Augsburg, West Germany

    M. Grötschel, M. Jünger & G. Reinelt

Authors
  1. M. Grötschel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Jünger
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Reinelt
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, Canada

    Ivan Rival

Rights and permissions

Reprints and permissions

Copyright information

© 1985 D. Reidel Publishing Company

About this chapter

Cite this chapter

Grötschel, M., Jünger, M., Reinelt, G. (1985). Acyclic Subdigraphs and Linear Orderings: Polytopes, Facets, and a Cutting Plane Algorithm. In: Rival, I. (eds) Graphs and Order. NATO ASI Series, vol 147. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5315-4_7

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-5315-4_7

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-8848-0

  • Online ISBN: 978-94-009-5315-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation