Decomposition of submodular functions

Abstract

A decomposition theory for submodular functions is described. Any such function is shown to have a unique decomposition consisting of indecomposable functions and certain highly decomposable functions, and the latter are completely characterized. Applications include decompositions of hypergraphs based on edge and vertex connectivity, the decomposition of matroids based on three-connectivity, the Gomory—Hu decomposition of flow networks, and Fujishige’s decomposition of symmetric submodular functions. Efficient decomposition algorithms are also discussed.

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References

  1. [1]

    R. E. Bixby andW. H. Cunningham, Matroids, graphs, and 3-connectivity,in: Bondy and Murty (eds.)Graph Theory and Related Topics, Academic Press, New York (1979), 91–103.

    Google Scholar 

  2. [2]

    R. E. Bixby, W. H. Cunningham andD. M. Topkis. The poset of a polymatroid vertex, extreme point,Report No. 82250—OR, Universität Bonn, 1982.

  3. [3]

    W. H. Cunningham, Decomposition of directed graphs,SIAM J. Algebraic and Discrete Methods 3 (1982), 214–228.

    MATH  MathSciNet  Article  Google Scholar 

  4. [4]

    W. H. Cunningham, Testing membership in matroid polyhedra,Report No. 81207—OR, Universität Bonn, 1981, to appear inJ. Combinatorial Th. Ser B.

  5. [5]

    W. H. Cunningham andJ. Edmonds, A combinatorial decomposition theory,Canad. J. Math. 32 (1980), 734–765.

    MATH  MathSciNet  Google Scholar 

  6. [6]

    W. H. Cunningham andJ. Edmonds, Decomposition of linear systems,in preparation.

  7. [7]

    J. Edmonds, Submodular functions, matroids, and certain polyhedra,in: R. K. Guy et al. (eds.)Combinatorial Structures, Gordon and Breach, New York (1970), 69–87.

    Google Scholar 

  8. [8]

    S. Fujishige, Canonical decompositions of symmetric submodular systems,in: Saito and Nishizeki (eds.)Graph Theory and Algorithms, Springer Verlag Lecture Notes in C. S.108 (1981), 53–64, also Discr. Applied Math.5 (1983), 175–190.

  9. [9]

    R. E. Gomory andT. C. Hu, Multi-terminal network flows, SIAM J. Appl. Math.9 (1961), 551–570.

    MATH  Article  MathSciNet  Google Scholar 

  10. [10]

    M. Grötschel, A. Schrijver andL. Lovász, The ellipsoid method and its consequences in combinatorial optimization,Combinatorica 1 (1981), 169–197.

    MATH  MathSciNet  Google Scholar 

  11. 11]

    L. Lovász,private communication, 1982.

  12. [12]

    J. M. Tan.Matroid 3-Connectivity, Thesis, Carleton University, 1981.

  13. [13]

    W. T. Tutte,Connectivity in Graphs, University of Toronto Press, 1966.

  14. [14]

    W. T. Tutte, Connectivity in matroids,Canad. J. Math. 18 (1966), 1301–1324.

    MATH  MathSciNet  Google Scholar 

  15. [15]

    L. A. Wolsey,private communication, 1981.

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Supported by Songerforschungsbereich 21 DFG, Institut für Operations Research Universität Bonn and by an N.S.E.R.C. of Canada operating grant.

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Cunningham, W.H. Decomposition of submodular functions. Combinatorica 3, 53–68 (1983). https://doi.org/10.1007/BF02579341

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AMS subject classification (1980)

  • 05 C 99
  • 05 B 35, 68 E 99