Advertisement

Mathematical Programming

, Volume 78, Issue 2, pp 195–217 | Cite as

Flows on hypergraphs

  • Riccardo Cambini
  • Giorgio Gallo
  • Maria Grazia Scutellà
Article

Abstract

We consider the capacitated minimum cost flow problem on directed hypergraphs. We define spanning hypertrees so generalizing the spanning tree of a standard graph, and show that, like in the standard and in the generalized minimum cost flow problems, a correspondence exists between bases and spanning hypertrees. Then, we show that, like for the network simplex algorithms for the standard and for the generalized minimum cost flow problems, most of the computations performed at each pivot operation have direct hypergraph interpretations.

Keywords

Flows Leontief flows Hypergraphs Simplex algorithm 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R.K. Ahuja, T.L. Magnanti and J.B. Orlin,Network Flows: Theory, Algorithms, and Applications (Prentice Hall, Englewood Cliffs, NJ, 1993).Google Scholar
  2. [2]
    G. Ausiello, A. D’Atri and D. Saccà, Graph algorithms for functional dependency manipulation,Journal of ACM 30 (1983) 752–766.zbMATHCrossRefGoogle Scholar
  3. [3]
    R. Cambini, G. Gallo and M.G. Scutellà, Minimum cost flows on hypergraphs, Technical Report TR-1/92, Dipartimento di Informatica, University of Pisa, Pisa, 1992.Google Scholar
  4. [4]
    G. Gallo, F. Licheri and M.G. Scutellà, The hypergraph simplex approach: Some experimental results,Ricerca Operativa ANNO XXVI, 78 (1996) 21–54.Google Scholar
  5. [5]
    G. Gallo, G. Longo, S. Nguyen and S. Pallottino, Directed hypergraphs and applications,Discrete Applied Mathematics 42 (1993) 177–201.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    G. Gallo and G. Rago, A hypergraph approach to logical inference for datalog formulae, Technical Report TR-28/90, Dipartimento di Informatica, University of Pisa, Pisa, 1990.Google Scholar
  7. [7]
    R.G. Jeroslow, R.K. Martin, R.R. Rardin and J. Wang, Gainfree Leontief substitution flow problems,Mathematical Programming 57 (1992) 375–414.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    J.L. Kennington and R.V. Helgason,Algorithms for Network Programming (Wiley/Interscience, New York, 1980) Chapter 7.zbMATHGoogle Scholar
  9. [9]
    G.J. Koehler, A.B. Whinston and G.P. Wright,Optimization over Leontief Substitution Systems (North-Holland, Amsterdam, 1975).zbMATHGoogle Scholar
  10. [10]
    E. Lawler,Combinatorial Optimization: Networks and Matroids (Holt, Rinehart and Winston, New York, 1976).zbMATHGoogle Scholar
  11. [11]
    A.F. Torres and J.D. Araoz, Combinatorial models for searching in knowledge bases,Mathematicas, Acta Cientifica Venezolana 39 (1988) 387–394.Google Scholar
  12. [12]
    A.F. Veinott Jr, Extreme points of Leontief substitution systems,Linear Algebra and Its Application 1 (1968) 181–194.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Mathematical Programming Society, Inc 1997

Authors and Affiliations

  • Riccardo Cambini
    • 1
  • Giorgio Gallo
    • 2
  • Maria Grazia Scutellà
    • 2
  1. 1.Dipartimento di Statistica e Matematica applicata all’EconomiaUniversità di PisaPisaItaly
  2. 2.Dipartimento di InformaticaUniversità di PisaPisaItaly

Personalised recommendations