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Min-Max Results in Combinatorial Optimization

  • A. Schrijver

Abstract

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]:

Keywords

Bipartite Graph Undirected Graph Submodular Function Perfect Graph Incidence Vector 
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References

  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

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