Min-Max Results in Combinatorial Optimization

  • A. Schrijver


Often the optimum of a combinatorial optimization problem is characterized by a min-max relation, asserting that the maximum value in one combinatorial optimization problem is equal to the minimum value in some other optimization problem. One of the best-known examples is the max-flow min-cut theorem of Ford and Fulkerson [1956] and Elias, Feinstein and Shannon [1956]:


Bipartite Graph Undirected Graph Submodular Function Perfect Graph Incidence Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1982]
    R. P. Anstee and M. Farber, Characterizations of totally balanced matrices, Research Report CORR 82–5, Faculty of Mathematics, University of Waterloo, Waterloo, Ont., 1982.Google Scholar
  2. [1981]
    E. Balas and N. Christofides, A restricted Lagrangean approach to the traveling salesman problem, Math. Programming 21 (1981) 19–46.MathSciNetMATHCrossRefGoogle Scholar
  3. [1950]
    H. B. Belck, Reguläre Faktoren von Graphen, J. Reine Angew. Math. 188 (1950) 228–252.MathSciNetMATHCrossRefGoogle Scholar
  4. [1958]
    C. Berge, Sur le couplage maximum d’un graphe, C. R. Acad. Sci. Paris 247 (1958) 258–259.MathSciNetMATHGoogle Scholar
  5. [1960]
    C. Berge, Les problèmes de coloration en théorie des graphes, Publ. Inst. Stat. Univ. Paris 9 (1960) 123–160.MathSciNetMATHGoogle Scholar
  6. [1961]
    C. Berge, Färbung von Graphen deren sämtliche bzw. ungerade Kreise starr sind (Zusammenfassung), Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, Math.-Natur. Reihe (1961) 114–115.Google Scholar
  7. [1962]
    C. Berge, Sur un conjecture relative au problème des codes optimaux, Com¬mun. 13ème Assemblée Gén. U.R.S.I., Tokyo, 1962.Google Scholar
  8. [1969]
    C. Berge, The rank of a family of sets and some applications to graph theory, in: Recent progress in combinatorics ( W. T. Tutte, ed.), Acad. Press, New York, 1969, pp. 246–257.Google Scholar
  9. [1970]
    C. Berge, Sur certain hypergraphes généralisant les graphes bipartis, in: Combinatorial theory and its applications ( P. Erdös, A. Rényi, and V. T. Sôs, eds.), North-Holland, Amsterdam, 1970, pp. 119–133.Google Scholar
  10. [1972]
    C. Berge, Balanced matrices, Math. Programming 2 (1972) 19–31.MathSciNetMATHCrossRefGoogle Scholar
  11. [1982]
    C. Berge and V. Chvâtal (eds.), Perfect graphs, to appear.Google Scholar
  12. [1970]
    C. Berge and M. Las Vergnas, Sur un théorème du type König pour hyper- graphes, in: Proc. Intern. Conf. on Comb. Math. (A. Gewirtz and L. Quintas, eds.), Ann. New York Acad. Sci. 175 (1970) 32–40.Google Scholar
  13. [1946]
    G. Birkhoff, Très observaciones sobre el algebra lineal, Rev. Univ Nac. Tucuman Ser. A 5 (1946) 147–148.MathSciNetMATHGoogle Scholar
  14. [1976]
    J. A. Bondy and U. S. R. Murty, Graph theory with applications, Macmillan, London, 1976.Google Scholar
  15. [1980]
    A. E. Brouwer and A. Kolen, A super-balanced hypergraph has a nest point, Report ZW 148/80, Math. Centrum, Amsterdam, 1980.Google Scholar
  16. [1982]
    K. Cameron, Polyhedral and algorithmic ramifications of antichains, Ph. D. thesis, University of Waterloo, Waterloo, Ont., 1982.Google Scholar
  17. [1975]
    Y. Chvâtal, On certain polytopes associated with graphs, J. Combinatorial Theory (B) 18 (1975) 138–154.MATHCrossRefGoogle Scholar
  18. [1981]
    V. Chvâtal, communication C.I.R.M. Marseille-Luminy, 1981.Google Scholar
  19. [1980]
    G. Cornuéjols and W. R. Pulleyblank, A matching problem with side conditions, Discrete Math. 29 (1980) 135–159.MathSciNetMATHCrossRefGoogle Scholar
  20. [1983]
    W. Cook and W. R. Pulleyblank, to appear.Google Scholar
  21. [1978]
    W. H. Cunningham and A. B. Marsh, A primal algorithm for optimal matching, Math. Programming Study 8 (1978) 50–72.MathSciNetGoogle Scholar
  22. [1951]
    G. B. Dantzig, Application of the simplex method to a transportation problem, in: Activity analysis of production and allocation ( T. C. Koopmans, ed.), J. Wiley, New York, 1951, pp. 359–373.Google Scholar
  23. [1979]
    R. W. Deming, Independence numbers of graphs - an extension of the König- Egervâry theorem, Discrete Math. 27 (1979) 23–33.MathSciNetMATHCrossRefGoogle Scholar
  24. [1971]
    M. A. H. Dempster, Two algorithms for the time-table problem, in: Combinatorial mathematics and its applications ( D. J. A. Welsh, ed.), Acad. Press, New York, 1971, pp. 63–85.Google Scholar
  25. [1950]
    R. P. Dilworth, A decomposition theorem for partially ordered sets, Ann. of Math. 51 (1950) 161–166.MathSciNetMATHCrossRefGoogle Scholar
  26. [1961]
    G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961) 71–76.MathSciNetMATHCrossRefGoogle Scholar
  27. [1965a]
    J. Edmonds, Minimum partition of a matroid into independent subsets, J. Res. Nat. Bur. Standards Sect. B69 (1965) 67–72.MathSciNetMATHGoogle Scholar
  28. [1965b]
    b] J. Edmonds, Lehman’s switching game and a theorem of Tutte and Nash-Williams, J. Res. Nat. Bur. Standards Sect. B69 (1965) 73–77.MathSciNetMATHGoogle Scholar
  29. [1965c]
    c] J. Edmonds, Paths, trees, and flowers, Canad. J. Math. 17 (1965) 449–467.MathSciNetMATHGoogle Scholar
  30. [1965d]
    J. Edmonds, Maximum matching and a polyhedron with 0,1-vertices, J. Res. Nat. Bur. Standards Sect. B69 (1965) 125–130.MathSciNetMATHGoogle Scholar
  31. [1967]
    J. Edmonds, An introduction to matching, mimeographed notes, Engineering Summer Conf., Univ. of Michigan, Ann Arbor, 1967.Google Scholar
  32. [1967a]
    a] J. Edmonds, Optimum branchings, J. Res. Nat. Bur. Standards Sect. B71 (1967) 233–240.MathSciNetMATHGoogle Scholar
  33. [1970]
    J. Edmonds, Submodular functions, matroids, and certain polyhedra, in: Combinatorial structures and their applications ( R. Guy, H. Hanani, N. Sauer and J. Schönheim, eds.), Gordon and Breach, New York, 1970, pp. 69–87.Google Scholar
  34. [1971]
    J. Edmonds, Matroids and the greedy algorithm, Math. Programming 1 (1971) 127–136.MathSciNetMATHCrossRefGoogle Scholar
  35. [1973]
    J. Edmonds, Edge-disjoint branchings, in: Combinatorial algorithms ( B. Rustin, ed.), Acad. Press, New York, 1973, pp. 91–96.Google Scholar
  36. [1979]
    J. Edmonds, Matroid intersection, Annals of Discrete Math. 4 (1979) 39–49.MathSciNetMATHCrossRefGoogle Scholar
  37. [1965]
    J. Edmonds and D. R. Fulkerson, Transversals and matroid partition, J. Res. Nat. Bur. Standards Sect. B69 (1965) 147–153.MathSciNetMATHGoogle Scholar
  38. [1970]
    J. Edmonds and D. R. Fulkerson, Bottleneck extrema, J. Combinatorial Theory 8 (1970) 299–306.MathSciNetMATHCrossRefGoogle Scholar
  39. [1977]
    J. Edmonds and R. Giles, A min-max relation for submodular functions on graphs, Annals of Discrete Math. 1 (1977) 185–204.MathSciNetCrossRefGoogle Scholar
  40. [1970]
    J. Edmonds and E. L. Johnson, Matching, a well-solved class of integer linear programs, in: Combinatorial structures and their applications ( R. Guy, H. Hanani, N. Sauer and J. Schonheim, eds.), Gordon and Breach, New York, 1970, pp. 89–92.Google Scholar
  41. [1973]
    J. Edmonds and E. L. Johnson, Matching, Euler tours and the Chinese postman, Math. Programming 5 (1973) 88–124.MathSciNetMATHCrossRefGoogle Scholar
  42. [1972]
    J. Edmonds and R. M. Karp, Theoretical improvements in algorithmic efficiency for network flow problems, J. ACM 19 (1972) 248–264.MATHCrossRefGoogle Scholar
  43. [1931]
    E. Egervary, Matrixok kombinatorius tulajdonsagairol, Mat. Fiz. Lapok 38 (1931) 16–28.MATHGoogle Scholar
  44. [1956]
    P. Elias, A. Feinstein and C. E. Shannon, A note on the maximum flow through a network, IRE Trans. Information Theory IT 2 (1956) 117–119.CrossRefGoogle Scholar
  45. [1956]
    L. R. Ford and D. R. Fulkerson, Maximum flow through a network, Canad. J. Math. 8 (1956) 399–404.MathSciNetMATHGoogle Scholar
  46. [1958]
    L. R. Ford and D. R. Fulkerson, Network flow and systems of distinct repre¬sentatives, Canad. J. Math. 10 (1958) 78–84.MathSciNetMATHGoogle Scholar
  47. [1962]
    L. R. Ford and D. R. Fulkerson, Flows in networks, Princeton Univ. Press, Princeton, N.J., 1962.Google Scholar
  48. [1979]
    A. Frank, Kernel systems of directed graphs, Acta Sci. Math. (Szeged) 41 (1979) 63–76.MATHGoogle Scholar
  49. [1980]
    A. Frank, On chain and antichain families of a partially ordered set, J. Combinatorial Theory (B) 29 (1980) 176–184.MATHCrossRefGoogle Scholar
  50. [1981]
    A. Frank, How to make a digraph strongly connected, Combinatorica 1 (1981) 145–153.MathSciNetMATHCrossRefGoogle Scholar
  51. [1912]
    G. Frobenius, Über Matrizen aus nicht negativen Elementen, Sitzber. Preuss. Akad. Wiss. (1912) 456–477.Google Scholar
  52. [1917]
    G. Frobenius, Über zerlegbare Determinanten, Sitzber. Preuss. Akad. Wiss. (1917) 274–277.Google Scholar
  53. [1956]
    D. R. Fulkerson, Note on Dilworth’s decomposition theorem for partially ordered sets, Proc. Amer. Math. Soc. 7 (1956) 701–702.MathSciNetMATHGoogle Scholar
  54. [1961]
    D. R. Fulkerson, An out-of-kilter method for minimal cost flow problems, SIAM J. Appl. Math. 9 (1961) 18–27.MATHGoogle Scholar
  55. [1968]
    D. R. Fulkerson, Networks, frames, and blocking systems, in: Mathematics of the decision sciences, part I (G. B. Dantzig and A. F. Veinott, eds.), Amer. Math. Soc., Providence, R. I., 1968, pp. 303–334.Google Scholar
  56. [1970]
    D. R. Fulkerson, Blocking polyhedra, in: Graph theory and its applications ( B. Harris, ed.), Acad. Press, New York, 1970, pp. 93–112.Google Scholar
  57. [1971]
    D. R. Fulkerson, Blocking and anti-blocking pairs of polyhedra, Math. Programming 1 (1971) 168–194.MathSciNetMATHCrossRefGoogle Scholar
  58. [1972]
    D. R. Fulkerson, Anti-blocking polyhedra, J. Combinatorial Theory (B) 12 (1972) 50–71.MathSciNetMATHCrossRefGoogle Scholar
  59. [1974]
    D. R. Fulkerson, Packing rooted directed cuts in a weighted directed graph, Math. Programming 6 (1974) 1–13.MathSciNetMATHCrossRefGoogle Scholar
  60. [1974]
    D. R. Fulkerson, A. J. Hoffman and R. Oppenheim, On balanced matrices, Math. Programming Study 1 (1974) 120–132.MathSciNetGoogle Scholar
  61. [91768]
    D. Gale, Optimal assignments in an ordered set: an application of matroid theory, J. Combinatorial Theory 4 (1968) 176–180.MathSciNetMATHCrossRefGoogle Scholar
  62. [1958]
    T. Gallai, Maximum-minimum Sätze über Graphen, Acta Math. Acad. Sci. Hungar. 9 (1958) 395–434.MathSciNetMATHCrossRefGoogle Scholar
  63. [1959]
    T. Gallai, Über extreme Punkt- und Kantenmengen, Ann. Univ. Sci. Budapest, Eötvos Sect. Math. 2 (1959) 133–138.MathSciNetMATHGoogle Scholar
  64. [1979]
    M. R. Garey and D. S. Johnson, Computers and intractability: a guide to the theory of NP-completeness, Freeman, San Francisco, 1979.MATHGoogle Scholar
  65. [1973]
    F. Gavril, Algorithms for a maximum clique and a maximum independent set of a circle graph, Networks 3 (1973) 261–273.MathSciNetMATHCrossRefGoogle Scholar
  66. [1974]
    F. Gavril, Algorithms on circular-arc graphs, Networks 4 (1974) 357–369.MathSciNetMATHCrossRefGoogle Scholar
  67. [1962]
    A. Ghouila-Houri, Caracterisation des matrices totalement unimodulaires, C. R. Acad. Sci. Paris 254 (1962) 1192–1194.MathSciNetMATHGoogle Scholar
  68. [1982a]
    R. Giles, Optimum matching forests I: special weights, Math. Programming 22 (1982) 1–11.MathSciNetMATHCrossRefGoogle Scholar
  69. [1982b]
    R. Giles, Optimum matching forests II: general weights, Math. Programming 22 (1982) 12–38.MathSciNetMATHCrossRefGoogle Scholar
  70. [1982c]
    R. Giles, Optimum matching forests III: facets of matching forest polyhedra, Math. Programming 22 (1982) 39–51.MathSciNetMATHCrossRefGoogle Scholar
  71. [1981]
    R. Giles and L. E. Trotter, On stable set polyhedra for 13-free graphs, J. Combinatorial Theory (B) 31 (1981) 313–326.MathSciNetMATHCrossRefGoogle Scholar
  72. [1980]
    M. C. Golumbic, Algorithmic graph theory and perfect graphs, Acad. Press, New York, 1980.MATHGoogle Scholar
  73. [1976]
    C. Greene, Some partitions associated with a partially ordered set, J. Combinatorial Theory (A) 20 (1976) 69–79.MathSciNetMATHCrossRefGoogle Scholar
  74. [1976]
    C. Greene and D. J. Kleitman, The structure of Sperner k-families, J. Combinatorial Theory (A) 20 (1976) 41–68.MathSciNetCrossRefGoogle Scholar
  75. [1981]
    V. P. Grishuhin, Polyhedra related to a lattice, Math. Programming 21 (1981) 70–89.MathSciNetMATHCrossRefGoogle Scholar
  76. [1981]
    M. Grötschel, L. Loväsz and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981) 169–197.MathSciNetMATHCrossRefGoogle Scholar
  77. [1981a]
    M. Grötschel, L. Loväsz and A. Schrijver, Polynomial algorithms for perfect graphs, Res. Report WP 81.176-OR, Inst. Oper. Research, Univ. Bonn, 1981.Google Scholar
  78. [1967]
    R. P. Gupta, A decomposition theorem for bipartite graphs, in: Theory of graphs ( P. Rosenstiehl, ed.), Gordon and Breach, New York, 1967, pp. 135–138.Google Scholar
  79. [1978]
    R. P. Gupta, An edge-colouring theorem for bipartite graphs with applications, Discrete Math. 23 (1978) 229–233.MathSciNetMATHGoogle Scholar
  80. [1981]
    E. Györi, A minimax theorem on intervals, preprint Math. Inst. Hung. Acad. Sci. No. 54/1981, Budapest, 1981.Google Scholar
  81. [1958]
    A. Hajnal and J. Suränyi, Über die Auflösung von Graphen in vollständigen Teilgraphen, Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 1 (1958) 113–121.MATHGoogle Scholar
  82. [1935]
    P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935) 26–30.CrossRefGoogle Scholar
  83. [1978]
    R. Hassin, On network flows, Ph. D. thesis, Yale Univ., Boston, 1978.Google Scholar
  84. [1960]
    A. J. Hoffman, Some recent applications of the theory of linear inequalities to extremal combinatorial analysis, in: Combinatorial analysis (R. E. Bellman and M. Hall, eds.), Amer. Math. Soc., Providence, R. I., 1960, pp. 113–127.Google Scholar
  85. [1982]
    A. J. Hoffman, A. W. J. Kolen and M. Sakarovitch, Totally-balanced and greedy matrices, Report BW, Math. Centrum, Amsterdam, 1982.Google Scholar
  86. [1956]
    A. J. Hoffman and J. B. Kruskal, Integral boundary points of convex polyhe- dra, in: Linear inequalities and related systems (H. W. Kuhn and A. W. Tucker, eds.), Ann. of Math. Studies 38, Princeton Univ. Press, Princeton, N.J., 1956, pp. 233–246.Google Scholar
  87. [1963]
    A. J. Hoffman and H. M. Markowitz, A note on shortest path, assignment, and transportation problems, Naval Res. Logist. Quart. 10 (1963) 375–380.MathSciNetMATHCrossRefGoogle Scholar
  88. [1977]
    A. J. Hoffman and D. E. Schwartz, On partitions of partially ordered sets, J. Combinatorial Theory (B) 23 (1977) 3–13.MathSciNetMATHCrossRefGoogle Scholar
  89. [1978]
    A. J. Hoffman and D. E. Schwartz, On lattice polyhedra, in: Combinatorics ( A. Hajnal and V. T. Sos, eds.), North-Holland, Amsterdam, 1978, pp. 593–598.Google Scholar
  90. [1981]
    W.-L. Hsu, Y. Ikura and G. L. Nemhauser, A polynomial algorithm for maximum weighted vertex packings on graphs without long odd cycles, Math. Programming 20 (1981) 225–232.MathSciNetMATHCrossRefGoogle Scholar
  91. [1963]
    T. C. Hu, Multicommodity network flows, Operations Res. 11 (1963) 344–360.MATHCrossRefGoogle Scholar
  92. [1973]
    T. C. Hu, Two-commodity cut packing problem, Discrete Math. 4 (1973) 108–109.MATHGoogle Scholar
  93. [1974]
    T. A. Jenkyns, Matchoids: a generalization of matchings and matroids, Ph. D. thesis, Univ. of Waterloo, Waterloo, Ont., 1974.Google Scholar
  94. [1979]
    A. V. Karzanov, On the minimal number of arcs of a digraph meeting all its directed cutsets, to appear.Google Scholar
  95. [1970]
    D. J. Kleitman, A. Martin-Löf, B. Rothschild and A. Whinston, A matching theorem for graphs, J. Combinatorial Theory 8 (1970) 104 - 114.MathSciNetMATHCrossRefGoogle Scholar
  96. [1915]
    D. König, Vonalrendszerek es determinänsok (Line-systems and determinants), Matematikai es Termeszettudomänyi Ertesitö 33 (1915) 221–229 (in Hungarian).Google Scholar
  97. [1916]
    D. König, Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre, Math. Ann. 77 (1916) 453–465.Google Scholar
  98. [1931]
    D. König, Graphok es matrixok, Mat. Fiz. Lapok 38 (1931) 116–119.MATHGoogle Scholar
  99. [1932]
    D. König, Über trennende Knotenpunkte in Graphen (nebst Anwendungen auf Determinanten und Matrizen), Acta. Lit. Sci. Sect. Sci. Math. (Szeged) 6 (1932–1934) 155–179.Google Scholar
  100. [1955]
    H. W. Kuhn, The Hungarian method for solving the assignment problem, Naval Res. Logist. Quart. 2 (1955) 83–97.CrossRefGoogle Scholar
  101. [1956]
    H. W. Kuhn, Variants of the Hungarian method for the assignment problem, Naval Res. Logist. Quart. 3 (1956) 253–258.CrossRefGoogle Scholar
  102. [1971]
    E. L. Lawler, Matroids with parity conditions: a new class of combinatorial optimization problems, Memorandum ERL-M334, Univ. of California, Berkeley, 1971.Google Scholar
  103. [1975]
    E. L. Lawler, Matroid intersection algorithms, Math. Programming 9 (1975) 31–56.MathSciNetMATHCrossRefGoogle Scholar
  104. 1976]
    E. L. Lawler, Combinatorial optimization: networks and matroids, Holt, Rine- hart and Winston, New York, 1976.Google Scholar
  105. [1982]
    E. L. Lawler and C. U. Martel, Computing maximal “polymatroidal” network flows, Math, of Oper. Research 7 (1982) 334–347MathSciNetMATHGoogle Scholar
  106. [1979]
    A. Lehman, On the width-length inequality, Math. Programming 16 (1979) 245–259.MathSciNetMATHCrossRefGoogle Scholar
  107. [1978]
    M. V. Lomonosov, On the systems of flows in a network, Probl. Per. Inf. 14 (1978)60–73 (in Russian); English translation: Problems of Inf. Transmission 14 (1978) 280–290.MathSciNetGoogle Scholar
  108. [1982]
    M. V. Lomonosov, Combinatorial approach to multi-flow problems, preprint. 1970 ] L. Lovász, Subgraphs with prescribed valencies, J. Combinatorial Theory 8 (1970) 391–416.Google Scholar
  109. [1972]
    L. Lovász, Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972) 253–267.MathSciNetMATHCrossRefGoogle Scholar
  110. [1974]
    L. Lovász, Minimax theorems for hypergraphs, in: Hypergraph seminar (C. Berge and D. Ray-Chaudhuri, eds.), Springer Lecture Notes in Mathematics 411, Springer, Berlin, 1974, pp. 111–126.Google Scholar
  111. [1975]
    L. Lovász, 2-Matchings and 2-covers of hypergraphs, Acta. Math. Acad. Sci. Hungar. 26 (1975) 433–444.MATHCrossRefGoogle Scholar
  112. [1976a]
    L. Lovász, On two minimax theorems in graph theory, J. Combinatorial Theory (B) 21 (1976) 96–103.MATHCrossRefGoogle Scholar
  113. [1976b]
    L. Lovász, On some connectivity properties of Eulerian graphs, Acta. Math. Acad. Sci. Hungar. 28 (1976) 129–138.MathSciNetMATHCrossRefGoogle Scholar
  114. [1977]
    L. Lovász, Certain duality principles in integer programming, Annals of Discrete Math. 1 (1977) 363–374.CrossRefGoogle Scholar
  115. [1979]
    L. Lovász, On the Shannon capacity of a graph, IEEE Trans. Inform. Theory IT 25 (1979) 1–7.MATHCrossRefGoogle Scholar
  116. [1979a]
    L. Lovász, Graph theory and integer programming, Annals of Discrete Math. 4 (1979) 141–158.MATHCrossRefGoogle Scholar
  117. [1980a]
    L. Lovász, Selecting independent lines from a family of lines in a space, Acta Sci. Math. (Szeged) 42 (1980) 121–131.MATHGoogle Scholar
  118. [1980b]
    L. Lovász, Matroid matching and some applications, J. Combinatorial Theory (B) 28 (1980) 208–236.MATHGoogle Scholar
  119. [1981a]
    L. Lovász, The matroid matching problem, in: Algebraic methods in graph theory ( L. Lovász and V. T. Sós, eds.), North-Holland, Amsterdam, 1981, pp. 495–517.Google Scholar
  120. [1981b]
    L. Lovász, Perfect graphs, in: More selected topics in graph theory (L. W. Beineke and R. J. Wilson, eds.), to appear.Google Scholar
  121. [1982]
    L. Lovász, Ear-decompositions of matching-covered graphs, preprint, 1982.Google Scholar
  122. [1982]
    A. Lubiw, T-free matrices, M. Sc. thesis, Univ. of Waterloo, Waterloo, Ont., 1982.Google Scholar
  123. [1976]
    C. L. Lucchesi, A minimax equality for directed graphs, Ph. D. thesis, Univ. of Waterloo, Waterloo, Ont., 1976.Google Scholar
  124. [1978]
    C. L. Lucchesi and D. H. Younger, A minimax relation for directed graphs, J. London Math. Soc. (2) 17 (1978) 369–374.MathSciNetMATHCrossRefGoogle Scholar
  125. [1978a]
    W. Mader, Uber die Maximalzahl kantendisjunkter A-Wege, Arch. Math. (Basel) 30 (1978) 325–336.MathSciNetMATHGoogle Scholar
  126. [1978b]
    W. Mader, Uber die Maximalzahl kreuzungsfreier H-Wege, Arch. Math. (Basel) 31 (1978) 387–402.MathSciNetGoogle Scholar
  127. [1979]
    A. B. Marsh, Matching algorithms, Ph. D. thesis, Johns Hopkins Univ., Baltimore, 1979.Google Scholar
  128. [1972]
    C. J. H. McDiarmid, The solution of a time-tabling problem, J. Inst. Maths. Appl. 9 (1972) 23–34.MathSciNetMATHCrossRefGoogle Scholar
  129. [1927]
    K. Menger, Zur allgemeinen Kurventheorie, Fund. Math. 10 (1927) 96–115.MATHGoogle Scholar
  130. [1976]
    H. Meyniel, On the perfect graph conjecture, Discrete Math. 16 (1976) 339–342.MathSciNetCrossRefGoogle Scholar
  131. [1960]
    G. J. Minty, Monotone networks, Proc. Roy. Soc. London Ser. A 257 (1960) 194–212.MathSciNetMATHCrossRefGoogle Scholar
  132. [1980]
    G. J. Minty, On maximal independent sets of vertices in a claw-free graph, J. Combinatorial Theory (B) 28 (1980) 284–304.MathSciNetMATHCrossRefGoogle Scholar
  133. [1971]
    L. Mirsky, Transversal theory, Acad. Press, London, 1971.MATHGoogle Scholar
  134. [1961]
    C. St. J. A. Nash-Williams, Edge-disjoint spanning trees of finite graphs, J. London Math. Soc. 36 (1961) 445–450.MathSciNetMATHCrossRefGoogle Scholar
  135. [1964]
    C. St. J. A. Nash-Williams, Decomposition of finite graphs into forests, J. London Math. Soc. 39 (1964) 12.MathSciNetMATHCrossRefGoogle Scholar
  136. [1953]
    J. von Neumann, A certain zero-sum two-person game equivalent to the optimum assignment problem, in: Contributions to the theory of games II (A. W. Tucker and H. W. Kuhn, eds.), Annals of Math. Studies 38, Princeton Univ. Press, Princeton, N.J., 1953, pp. 5–12.Google Scholar
  137. [1981]
    H. Okamura and P. D. Seymour, Multicommodity flows in planar graphs, J. Combinatorial Theory (B) 31 (1981) 75–81.MathSciNetMATHCrossRefGoogle Scholar
  138. [1956]
    A. Orden, The transshipment problem, Manag. Sci. 2 (1956) 276–285.MathSciNetMATHGoogle Scholar
  139. [1975]
    M. Padberg, Characterisation of totally unimodular, balanced and perfect matrices, in: Combinatorial programming: methods and applications ( B. Roy, ed.), Reidel, Dordrecht (Holland), 1975, pp. 275–284.Google Scholar
  140. [1982]
    M. W. Padberg and M. R. Rao, Odd minimum cut-sets and b-matchings, Math, of Oper. Res. 7 (1982) 67–80.MathSciNetMATHGoogle Scholar
  141. [1982]
    C. H. Papadimitriou and K. Steiglitz, Combinatorial optimization: algorithms and complexity, Prentice-Hall, Englewood Cliffs, N.J., 1982.MATHGoogle Scholar
  142. [1976]
    B. A. Papernov, Feasibility of multicommodity flows, in: Studies in Discrete Optimization ( A. A. Fridman, ed.), Izdat. “Nauka”, Moscow, 1976, pp. 230–261 (in Russian).Google Scholar
  143. [1968]
    H. Perfect, Applications of Menger’s graph theorem, J. Math. Analysis Appl. 22 (1968) 96–111.MathSciNetMATHCrossRefGoogle Scholar
  144. [1973]
    W. R. Pulleyblank, Faces of matching polyhedra, Ph. D. thesis, Univ. of Waterloo, Waterloo, Ont., 1973.Google Scholar
  145. [1980]
    W. R. Pulleyblank, Dual integrality in b-matching problems, Math. Programming Study 12 (1980) 176–196.MathSciNetMATHGoogle Scholar
  146. [1983]
    W. R. Pulleyblank, Polyhedral combinatorics, this volume.Google Scholar
  147. [1957]
    R. Rado, A note on independence functions, Proc. London Math. Soc. 7 (1957) 300–320.MathSciNetMATHGoogle Scholar
  148. [1966a]
    B. Rothschild and A. Whinston, On two-commodity network flows, Operations Res. 14 (1966) 377–387.MathSciNetMATHCrossRefGoogle Scholar
  149. [1966b]
    B. Rothschild and A. Whinston, Feasibility of two-commodity network flows, Operations Res. 14 (1966) 1121–1129.MathSciNetMATHCrossRefGoogle Scholar
  150. [1978]
    N. Sbihi, Étude des stables dans les graphes sans étoile, M. Sc. thesis, Univ. Sci. et Méd. Grenoble, 1978.Google Scholar
  151. [1981]
    N. Sbihi and J. P. Uhry, A class of h-perfect graphs, Rapport de Rech. No. 236, IRMA, Grenoble, 1981.Google Scholar
  152. [1980]
    A. Schrijver, A counterexemple to a conjecture of Edmonds and Giles, Discrete Math. 32 (1980) 213–214.MathSciNetMATHGoogle Scholar
  153. [1981]
    A. Schrijver, Short proofs on the matching polyhedron, Rapport AE 17/81, Inst. Act. & Econ., Univ. van Amsterdam, Amsterdam, 1981 ( J. Combinatorial Theory (B), to appear).Google Scholar
  154. [1982a]
    a] A. Schrijver, Min-max relations for directed graphs, Annals of Discrete Math. 16 (1982) 261–280.MathSciNetMATHGoogle Scholar
  155. [1982b]
    A. Schrijver, Proving total dual integrality with cross-free families - a general framework, Report AE 5/82, Inst. Act. & Econ., Univ. van Amsterdam, Amsterdam, 1982 ( Math. Programming, to appear).Google Scholar
  156. [1982c]
    A. Schrijver, Total dual integrality from directed graphs, crossing families, and sub- and supermodular functions, Proc. Waterloo 1982, to appear.Google Scholar
  157. [1983a]
    a] A. Schrijver, Packing and covering of crossing families of cuts, Report AE 1/ 83, Univ. van Amsterdam, Amsterdam, 1983. 1983Google Scholar
  158. [1983b]
    b] A. Schrijver, Supermodular colourings, Report AE 4/83, Univ. van Amsterdam, Amsterdam, 1983.Google Scholar
  159. [1977]
    A. Schrijver and P. D. Seymour, A proof of total dual integrality of matching polyhedra, Report ZN 79/77, Math. Centrum, Amsterdam, 1977.Google Scholar
  160. [1979]
    A. Schrijver and P. D. Seymour, Solution of two fractional packing problems of Lovasz, Discrete Math. 26 (1979) 177–184.MathSciNetMATHCrossRefGoogle Scholar
  161. [1977]
    P. D. Seymour, The matroids with the max-flow min-cut property, J. Combina¬torial Theory (B) 23 (1977) 189–222.MathSciNetMATHCrossRefGoogle Scholar
  162. [1978a]
    a] P. D. Seymour, A two-commodity cut theorem, Discrete Math. 23 (1978) 177–181.MathSciNetMATHCrossRefGoogle Scholar
  163. [1978b]
    P. D. Seymour, Sums of circuits, in: Graph theory and related topics ( J. A. Bondy and U. S. R. Murty, eds.), Acad. Press, New York, 1978, pp. 341–355.Google Scholar
  164. [1979a]
    P. D. Seymour, On multi-colourings of cubic graphs, and conjectures of Fulkerson and Tutte, Proc. London Math. Soc. (3) 38 (1979) 423–460.MathSciNetMATHCrossRefGoogle Scholar
  165. [1979b]
    b] P. D. Seymour, A short proof of the two-commodity flow theorem, J. Combinatorial Theory (B) 26 (1979) 370–371.MathSciNetMATHCrossRefGoogle Scholar
  166. [1980]
    P. D. Seymour, Four-terminus flows, Networks 10 (1980) 79–86.MathSciNetMATHCrossRefGoogle Scholar
  167. [1981a]
    P. D. Seymour, On odd cuts and plane multicommodity flows, Proc. London Math. Soc. (3) 42 (1981) 178–192.MathSciNetMATHCrossRefGoogle Scholar
  168. [1981b]
    P. D. Seymour, Matroids and multicommodity flows, Europ. J. Comb. 2 (1981) 257–290.MathSciNetMATHGoogle Scholar
  169. [1979]
    F. Sterboul, A characterization of the graphs in which the transversal number equals the matching number, J. Combinatorial Theory (B) 27 (1979) 228–229.MathSciNetMATHCrossRefGoogle Scholar
  170. [1970]
    J. Stoer and C. Witzgall, Convexity and optimization in finite dimensions IGoogle Scholar
  171. Springer, Berlin, 1970.Google Scholar
  172. [1947]
    W. T. Tutte, The factorization of linear graphs, J. London Math. Soc. 22 (1947) 107–111.MathSciNetMATHCrossRefGoogle Scholar
  173. [1952]
    W. T. Tutte, The factors of graphs, Canad. J. Math. 4 (1952) 314–328.MathSciNetMATHGoogle Scholar
  174. [1953]
    W. T. Tutte, The 1-factors of oriented graphs, Proc. Amer. Math. Soc. 4 (1953) 922–931.MathSciNetMATHGoogle Scholar
  175. [1954]
    W. T. Tutte, A short proof of the factor theorem for finite graphs, Canad. J. Math. 6 (1954) 347–352.MathSciNetMATHGoogle Scholar
  176. [1961]
    W. T. Tutte, On the problem of decomposing a graph into n connected factors, J. London Math. Soc. 36 (1961) 221–230.MathSciNetMATHCrossRefGoogle Scholar
  177. [1981]
    W. T. Tutte, Graph factors, Combinatorica 1 (1981) 79–97.MathSciNetMATHGoogle Scholar
  178. [1964]
    V. G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz. 3 (1964) 25–30 (in Russian).MathSciNetGoogle Scholar
  179. [1937]
    B. L. van der Waerden, Moderne Algebra, Springer, Berlin, 1937.Google Scholar
  180. [1970]
    D. J. A. Welsh, On matroid theorems of Edmonds and Rado, J. London Math. Soc. 2 (1970) 251–256.MathSciNetMATHCrossRefGoogle Scholar
  181. [1976]
    D. J. A. Welsh, Matroid theory, Acad. Press, London, 1976.MATHGoogle Scholar
  182. [1970]
    D. de Werra, On some combinatorial problems arising in scheduling, Canad. Oper. Res. Soc. J. 8 (1970) 165–175.Google Scholar
  183. [1972]
    D. de Werra, Decomposition of bipartite multigraphs into matchings, Zeitschr. Oper. Res. 16 (1972) 85–90.CrossRefGoogle Scholar
  184. [1935]
    H. Whitney, On the abstract properties of linear independence, Amer. J. Math. 57 (1935) 509–533.MathSciNetCrossRefGoogle Scholar
  185. [1972]
    R. J. Wilson, Introduction to graph theory, Oliver and Boyd, Edinburgh, 1972.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • A. Schrijver
    • 1
  1. 1.Instituut voor Actuariaat en EconometrieUniversiteit van AmsterdamAmsterdamThe Netherlands

Personalised recommendations