The Schrijver system of odd join polyhedra


Graphs for which the set oft-joins andt-cuts has “the max-flow-min-cut property”, i.e. for which the minimal defining system of thet-join polyhedron is totally dual integral, have been characterized by Seymour. An extension of this problem isto characterize the (uniquely existing) minimal totally dual integral defining system (Schrijver-system) of an arbitrary t-join polyhedron. This problem is solved in the present paper. The main idea is to uset-borders, a generalization oft-cuts, to obtain an integer minimax formula for the cardinality of a minimumt-join. (At-border is the set of edges joining different classes of a partition of the vertex set intot-odd sets.) It turns out that the (uniquely existing) “strongest minimax theorem” involves just this notion.

This is a preview of subscription content, access via your institution.


  1. [1]

    W. Cook, A minimal totally dual integral defining system for theb-matching polyhedron,SIAM Journal on Alg. and Disc. Methods,4 (1983), 212–220.

    Google Scholar 

  2. [2]

    W. Cook, On Some Aspects of Totally Dual Integral Systems,Ph. D. Thesis, University of Waterloo, Canada, 1983.

    Google Scholar 

  3. [3]

    W. Cook, A note on matchings and separability,Report 84315-OR, Institut für Ökonometrie und Operations Research, Bonn, W. Germany, 1984.

    Google Scholar 

  4. [4]

    W. Cook andW. R. Pulleyblank, Linear Systems for Constrained Matching Problems,Report No. 84323-OR, Institut für Ökonometrie und Operations Research, Bonn, W. Germany, 1984.

    Google Scholar 

  5. [5]

    W. H. Cunningham andA. B. Marsh III, A primal algorithm for optimum matching,Math. Prog. Study. 8 (1978), 50–72.

    Google Scholar 

  6. [6]

    J. Edmonds andE. L. Johnson, Matching: a well-solved class of integer linear programs, in:R. Guy, H. Hanani, N. Sauer, andJ. Schonheim (eds..Combinatorial Structures and Their Applications (Gordon and Breach, New York, 1970), 89–92.

    Google Scholar 

  7. [7]

    J. Edmonds andE. L. Johnson, Matching, Euler tours and the Chinese postman,Math Programming,5 (1973), 88–124.

    Google Scholar 

  8. [8]

    J. Edmonds, L. Lovász andW. R. Pulleyblank, Brick Decompositions and the Matching Rank of Graphs,Combinatorica,2 (1982), 247–274.

    Google Scholar 

  9. [9]

    A. Frank, A. Sebő andÉ. Tardos, Covering Directed and Odd Cuts,Mathematical Programming Study.22 (1984), 99–112.

    Google Scholar 

  10. [10]

    A.Gerards, A.Schrijver, Signed Graphs—Regular Matroids—Grafts,Tilburg University, Department of Econometrics. Tilburg.

  11. [11]

    A.Gerards, A.Sebő, Total dual integrality implies local strong unimodularity,Mathematical Programming.

  12. [12]

    A.Gerards,private communication.

  13. [13]

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

    Google Scholar 

  14. [14]

    E.Korach, Packing ofT-cuts and Other Aspects of Dual Integrality,Ph. D. Thesis, Waterloo University. 1982.

  15. [15]

    L. Lovász, On the structure of factorizable graphs,Acta Math. Acad. Sci. Hung.,23 (1972), 179–195.

    Google Scholar 

  16. [16]

    L. Lovász, 2-matchings and 2-covers of hypergraphs,Acta Mathematica Academiae Scientiarum Hungaricae.26 (1975), 433–444.

    Google Scholar 

  17. [17]

    L.Lovász and M. D.Plummer, On bicritical graphs, Infinite and Finite Sets,Colloqu. Math. Soc. J. Bolyai 10,Budapest (A. Hajnal, R. Rado and V. T. Sós et al (eds.)), (1975), 1051–1079.

  18. [18]

    L.Lovász and M. D.Plummer,Matching Theory, Akadémiai Kiadó, 1985.

  19. [19]

    Mei Gu Guan, Graphic programming using odd or even points,Chinese Math.,1 (1962), 273–277.

    Google Scholar 

  20. [20]

    W. R. Pulleyblank, Dual integrality inb-matching problems,Math. Prog. Study,12 (1980), 176–196.

    Google Scholar 

  21. [21]

    W. R. Pulleyblank, Total dual integrality andb-matchings,Operations Research Letters,1 (1981), 28–30.

    Google Scholar 

  22. [22]

    A. Schrijver, On Cutting Planes,Annals Discrete Math.,9 (1980), 291–296.

    Google Scholar 

  23. [23]

    A. Schrijver, On total dual integrality,Lin. Alg. and its Appl.,38 (1981), 27–32.

    Google Scholar 

  24. [24]

    A. Schrijver, Min-max results in combinatorial optimization, in:A. Bachern, M. Grötschel, andB. Karte (eds..Mathematical Programming —The State of the Art (Springer Verlag, Heidelberg, 1983), 439–500.

    Google Scholar 

  25. [25]

    A. Schrijver,Theory of linear and integer programming, Wiley, Chichester, (1986).

    Google Scholar 

  26. [26]

    A.Sebő, Undirected Distances and the Postman Structure of Graphs,to appear in Journal of Combinatorial Theory, 1988.

  27. [27]

    A. Sebő, Finding thet-join Structure of Graphs,Mathematical Programming,36 (1986), 123–134.

    Google Scholar 

  28. [28]

    A.Sebő, The Chinese Postman Problem: Algorithms, Structure and Applications,SZTAKI Working Paper, MO/61, (1985).

  29. [29]

    A. Sebő, A quick proof of Seymour's theorem ont-joins,Discrete Mathematics,64 (1987), 101–103.

    Google Scholar 

  30. [30]

    A.Sebő, C. Sc. thesis(in Hungarian), august 1987.

  31. [31]

    P. D. Seymour, The matroids with the Max-Flow-Min-Cut Property,J. Comb. Theory, Series B.23 (1977), 189–222.

    Google Scholar 

  32. [32]

    P. D. Seymour, On odd cuts and plane multicommodity flows,Proc. London Math. Soc., 3/42 (1981), 178–192.

    Google Scholar 

  33. [33]

    É.Tardos,private communication.

Download references

Author information



Additional information

On leave from Eötvös Loránd University and Computer and Automation Institute, Budapest.

Supported by Sonderforschungsbereich 303 (DFG), Institut für Operations Research, Universität Bonn, W. Germany

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sebő, A. The Schrijver system of odd join polyhedra. Combinatorica 8, 103–116 (1988).

Download citation

AMS subject classification (1980)

  • 90 C 10
  • 05 C 70