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Min-Max Results in Combinatorial Optimization

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Mathematical Programming The State of the Art

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]:

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References

  1. 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. E. Balas and N. Christofides, A restricted Lagrangean approach to the traveling salesman problem, Math. Programming 21 (1981) 19–46.

    Article  MathSciNet  MATH  Google Scholar 

  3. H. B. Belck, Reguläre Faktoren von Graphen, J. Reine Angew. Math. 188 (1950) 228–252.

    Article  MathSciNet  MATH  Google Scholar 

  4. C. Berge, Sur le couplage maximum d’un graphe, C. R. Acad. Sci. Paris 247 (1958) 258–259.

    MathSciNet  MATH  Google Scholar 

  5. C. Berge, Les problèmes de coloration en théorie des graphes, Publ. Inst. Stat. Univ. Paris 9 (1960) 123–160.

    MathSciNet  MATH  Google Scholar 

  6. 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. 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. 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. 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. C. Berge, Balanced matrices, Math. Programming 2 (1972) 19–31.

    Article  MathSciNet  MATH  Google Scholar 

  11. C. Berge and V. Chvâtal (eds.), Perfect graphs, to appear.

    Google Scholar 

  12. 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. G. Birkhoff, Très observaciones sobre el algebra lineal, Rev. Univ Nac. Tucuman Ser. A 5 (1946) 147–148.

    MathSciNet  MATH  Google Scholar 

  14. J. A. Bondy and U. S. R. Murty, Graph theory with applications, Macmillan, London, 1976.

    Google Scholar 

  15. 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. K. Cameron, Polyhedral and algorithmic ramifications of antichains, Ph. D. thesis, University of Waterloo, Waterloo, Ont., 1982.

    Google Scholar 

  17. Y. Chvâtal, On certain polytopes associated with graphs, J. Combinatorial Theory (B) 18 (1975) 138–154.

    Article  MATH  Google Scholar 

  18. V. Chvâtal, communication C.I.R.M. Marseille-Luminy, 1981.

    Google Scholar 

  19. G. Cornuéjols and W. R. Pulleyblank, A matching problem with side conditions, Discrete Math. 29 (1980) 135–159.

    Article  MathSciNet  MATH  Google Scholar 

  20. W. Cook and W. R. Pulleyblank, to appear.

    Google Scholar 

  21. W. H. Cunningham and A. B. Marsh, A primal algorithm for optimal matching, Math. Programming Study 8 (1978) 50–72.

    MathSciNet  Google Scholar 

  22. 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. R. W. Deming, Independence numbers of graphs - an extension of the König- Egervâry theorem, Discrete Math. 27 (1979) 23–33.

    Article  MathSciNet  MATH  Google Scholar 

  24. 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. R. P. Dilworth, A decomposition theorem for partially ordered sets, Ann. of Math. 51 (1950) 161–166.

    Article  MathSciNet  MATH  Google Scholar 

  26. G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961) 71–76.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Edmonds, Minimum partition of a matroid into independent subsets, J. Res. Nat. Bur. Standards Sect. B69 (1965) 67–72.

    MathSciNet  MATH  Google Scholar 

  28. 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.

    MathSciNet  MATH  Google Scholar 

  29. c] J. Edmonds, Paths, trees, and flowers, Canad. J. Math. 17 (1965) 449–467.

    MathSciNet  MATH  Google Scholar 

  30. J. Edmonds, Maximum matching and a polyhedron with 0,1-vertices, J. Res. Nat. Bur. Standards Sect. B69 (1965) 125–130.

    MathSciNet  MATH  Google Scholar 

  31. J. Edmonds, An introduction to matching, mimeographed notes, Engineering Summer Conf., Univ. of Michigan, Ann Arbor, 1967.

    Google Scholar 

  32. a] J. Edmonds, Optimum branchings, J. Res. Nat. Bur. Standards Sect. B71 (1967) 233–240.

    MathSciNet  MATH  Google Scholar 

  33. 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. J. Edmonds, Matroids and the greedy algorithm, Math. Programming 1 (1971) 127–136.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Edmonds, Edge-disjoint branchings, in: Combinatorial algorithms ( B. Rustin, ed.), Acad. Press, New York, 1973, pp. 91–96.

    Google Scholar 

  36. J. Edmonds, Matroid intersection, Annals of Discrete Math. 4 (1979) 39–49.

    Article  MathSciNet  MATH  Google Scholar 

  37. J. Edmonds and D. R. Fulkerson, Transversals and matroid partition, J. Res. Nat. Bur. Standards Sect. B69 (1965) 147–153.

    MathSciNet  MATH  Google Scholar 

  38. J. Edmonds and D. R. Fulkerson, Bottleneck extrema, J. Combinatorial Theory 8 (1970) 299–306.

    Article  MathSciNet  MATH  Google Scholar 

  39. J. Edmonds and R. Giles, A min-max relation for submodular functions on graphs, Annals of Discrete Math. 1 (1977) 185–204.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. J. Edmonds and R. M. Karp, Theoretical improvements in algorithmic efficiency for network flow problems, J. ACM 19 (1972) 248–264.

    Article  MATH  Google Scholar 

  43. E. Egervary, Matrixok kombinatorius tulajdonsagairol, Mat. Fiz. Lapok 38 (1931) 16–28.

    MATH  Google Scholar 

  44. 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.

    Article  Google Scholar 

  45. L. R. Ford and D. R. Fulkerson, Maximum flow through a network, Canad. J. Math. 8 (1956) 399–404.

    MathSciNet  MATH  Google Scholar 

  46. L. R. Ford and D. R. Fulkerson, Network flow and systems of distinct repre¬sentatives, Canad. J. Math. 10 (1958) 78–84.

    MathSciNet  MATH  Google Scholar 

  47. L. R. Ford and D. R. Fulkerson, Flows in networks, Princeton Univ. Press, Princeton, N.J., 1962.

    Google Scholar 

  48. A. Frank, Kernel systems of directed graphs, Acta Sci. Math. (Szeged) 41 (1979) 63–76.

    MATH  Google Scholar 

  49. A. Frank, On chain and antichain families of a partially ordered set, J. Combinatorial Theory (B) 29 (1980) 176–184.

    Article  MATH  Google Scholar 

  50. A. Frank, How to make a digraph strongly connected, Combinatorica 1 (1981) 145–153.

    Article  MathSciNet  MATH  Google Scholar 

  51. G. Frobenius, Über Matrizen aus nicht negativen Elementen, Sitzber. Preuss. Akad. Wiss. (1912) 456–477.

    Google Scholar 

  52. G. Frobenius, Über zerlegbare Determinanten, Sitzber. Preuss. Akad. Wiss. (1917) 274–277.

    Google Scholar 

  53. D. R. Fulkerson, Note on Dilworth’s decomposition theorem for partially ordered sets, Proc. Amer. Math. Soc. 7 (1956) 701–702.

    MathSciNet  MATH  Google Scholar 

  54. D. R. Fulkerson, An out-of-kilter method for minimal cost flow problems, SIAM J. Appl. Math. 9 (1961) 18–27.

    MATH  Google Scholar 

  55. 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. 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. D. R. Fulkerson, Blocking and anti-blocking pairs of polyhedra, Math. Programming 1 (1971) 168–194.

    Article  MathSciNet  MATH  Google Scholar 

  58. D. R. Fulkerson, Anti-blocking polyhedra, J. Combinatorial Theory (B) 12 (1972) 50–71.

    Article  MathSciNet  MATH  Google Scholar 

  59. D. R. Fulkerson, Packing rooted directed cuts in a weighted directed graph, Math. Programming 6 (1974) 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  60. D. R. Fulkerson, A. J. Hoffman and R. Oppenheim, On balanced matrices, Math. Programming Study 1 (1974) 120–132.

    MathSciNet  Google Scholar 

  61. D. Gale, Optimal assignments in an ordered set: an application of matroid theory, J. Combinatorial Theory 4 (1968) 176–180.

    Article  MathSciNet  MATH  Google Scholar 

  62. T. Gallai, Maximum-minimum Sätze über Graphen, Acta Math. Acad. Sci. Hungar. 9 (1958) 395–434.

    Article  MathSciNet  MATH  Google Scholar 

  63. T. Gallai, Über extreme Punkt- und Kantenmengen, Ann. Univ. Sci. Budapest, Eötvos Sect. Math. 2 (1959) 133–138.

    MathSciNet  MATH  Google Scholar 

  64. M. R. Garey and D. S. Johnson, Computers and intractability: a guide to the theory of NP-completeness, Freeman, San Francisco, 1979.

    MATH  Google Scholar 

  65. F. Gavril, Algorithms for a maximum clique and a maximum independent set of a circle graph, Networks 3 (1973) 261–273.

    Article  MathSciNet  MATH  Google Scholar 

  66. F. Gavril, Algorithms on circular-arc graphs, Networks 4 (1974) 357–369.

    Article  MathSciNet  MATH  Google Scholar 

  67. A. Ghouila-Houri, Caracterisation des matrices totalement unimodulaires, C. R. Acad. Sci. Paris 254 (1962) 1192–1194.

    MathSciNet  MATH  Google Scholar 

  68. R. Giles, Optimum matching forests I: special weights, Math. Programming 22 (1982) 1–11.

    Article  MathSciNet  MATH  Google Scholar 

  69. R. Giles, Optimum matching forests II: general weights, Math. Programming 22 (1982) 12–38.

    Article  MathSciNet  MATH  Google Scholar 

  70. R. Giles, Optimum matching forests III: facets of matching forest polyhedra, Math. Programming 22 (1982) 39–51.

    Article  MathSciNet  MATH  Google Scholar 

  71. R. Giles and L. E. Trotter, On stable set polyhedra for 13-free graphs, J. Combinatorial Theory (B) 31 (1981) 313–326.

    Article  MathSciNet  MATH  Google Scholar 

  72. M. C. Golumbic, Algorithmic graph theory and perfect graphs, Acad. Press, New York, 1980.

    MATH  Google Scholar 

  73. C. Greene, Some partitions associated with a partially ordered set, J. Combinatorial Theory (A) 20 (1976) 69–79.

    Article  MathSciNet  MATH  Google Scholar 

  74. C. Greene and D. J. Kleitman, The structure of Sperner k-families, J. Combinatorial Theory (A) 20 (1976) 41–68.

    Article  MathSciNet  Google Scholar 

  75. V. P. Grishuhin, Polyhedra related to a lattice, Math. Programming 21 (1981) 70–89.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  77. 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. 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. R. P. Gupta, An edge-colouring theorem for bipartite graphs with applications, Discrete Math. 23 (1978) 229–233.

    MathSciNet  MATH  Google Scholar 

  80. E. Györi, A minimax theorem on intervals, preprint Math. Inst. Hung. Acad. Sci. No. 54/1981, Budapest, 1981.

    Google Scholar 

  81. A. Hajnal and J. Suränyi, Über die Auflösung von Graphen in vollständigen Teilgraphen, Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 1 (1958) 113–121.

    MATH  Google Scholar 

  82. P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935) 26–30.

    Article  Google Scholar 

  83. R. Hassin, On network flows, Ph. D. thesis, Yale Univ., Boston, 1978.

    Google Scholar 

  84. 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. A. J. Hoffman, A. W. J. Kolen and M. Sakarovitch, Totally-balanced and greedy matrices, Report BW, Math. Centrum, Amsterdam, 1982.

    Google Scholar 

  86. 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. A. J. Hoffman and H. M. Markowitz, A note on shortest path, assignment, and transportation problems, Naval Res. Logist. Quart. 10 (1963) 375–380.

    Article  MathSciNet  MATH  Google Scholar 

  88. A. J. Hoffman and D. E. Schwartz, On partitions of partially ordered sets, J. Combinatorial Theory (B) 23 (1977) 3–13.

    Article  MathSciNet  MATH  Google Scholar 

  89. 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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  91. T. C. Hu, Multicommodity network flows, Operations Res. 11 (1963) 344–360.

    Article  MATH  Google Scholar 

  92. T. C. Hu, Two-commodity cut packing problem, Discrete Math. 4 (1973) 108–109.

    MATH  Google Scholar 

  93. T. A. Jenkyns, Matchoids: a generalization of matchings and matroids, Ph. D. thesis, Univ. of Waterloo, Waterloo, Ont., 1974.

    Google Scholar 

  94. A. V. Karzanov, On the minimal number of arcs of a digraph meeting all its directed cutsets, to appear.

    Google Scholar 

  95. D. J. Kleitman, A. Martin-Löf, B. Rothschild and A. Whinston, A matching theorem for graphs, J. Combinatorial Theory 8 (1970) 104 - 114.

    Article  MathSciNet  MATH  Google Scholar 

  96. D. König, Vonalrendszerek es determinänsok (Line-systems and determinants), Matematikai es Termeszettudomänyi Ertesitö 33 (1915) 221–229 (in Hungarian).

    Google Scholar 

  97. D. König, Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre, Math. Ann. 77 (1916) 453–465.

    Google Scholar 

  98. D. König, Graphok es matrixok, Mat. Fiz. Lapok 38 (1931) 116–119.

    MATH  Google Scholar 

  99. 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. H. W. Kuhn, The Hungarian method for solving the assignment problem, Naval Res. Logist. Quart. 2 (1955) 83–97.

    Article  Google Scholar 

  101. H. W. Kuhn, Variants of the Hungarian method for the assignment problem, Naval Res. Logist. Quart. 3 (1956) 253–258.

    Article  Google Scholar 

  102. 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. E. L. Lawler, Matroid intersection algorithms, Math. Programming 9 (1975) 31–56.

    Article  MathSciNet  MATH  Google Scholar 

  104. E. L. Lawler, Combinatorial optimization: networks and matroids, Holt, Rine- hart and Winston, New York, 1976.

    Google Scholar 

  105. E. L. Lawler and C. U. Martel, Computing maximal “polymatroidal” network flows, Math, of Oper. Research 7 (1982) 334–347

    MathSciNet  MATH  Google Scholar 

  106. A. Lehman, On the width-length inequality, Math. Programming 16 (1979) 245–259.

    Article  MathSciNet  MATH  Google Scholar 

  107. 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.

    MathSciNet  Google Scholar 

  108. 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. L. Lovász, Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972) 253–267.

    Article  MathSciNet  MATH  Google Scholar 

  110. 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. L. Lovász, 2-Matchings and 2-covers of hypergraphs, Acta. Math. Acad. Sci. Hungar. 26 (1975) 433–444.

    Article  MATH  Google Scholar 

  112. L. Lovász, On two minimax theorems in graph theory, J. Combinatorial Theory (B) 21 (1976) 96–103.

    Article  MATH  Google Scholar 

  113. L. Lovász, On some connectivity properties of Eulerian graphs, Acta. Math. Acad. Sci. Hungar. 28 (1976) 129–138.

    Article  MathSciNet  MATH  Google Scholar 

  114. L. Lovász, Certain duality principles in integer programming, Annals of Discrete Math. 1 (1977) 363–374.

    Article  Google Scholar 

  115. L. Lovász, On the Shannon capacity of a graph, IEEE Trans. Inform. Theory IT 25 (1979) 1–7.

    Article  MATH  Google Scholar 

  116. L. Lovász, Graph theory and integer programming, Annals of Discrete Math. 4 (1979) 141–158.

    Article  MATH  Google Scholar 

  117. L. Lovász, Selecting independent lines from a family of lines in a space, Acta Sci. Math. (Szeged) 42 (1980) 121–131.

    MATH  Google Scholar 

  118. L. Lovász, Matroid matching and some applications, J. Combinatorial Theory (B) 28 (1980) 208–236.

    MATH  Google Scholar 

  119. 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. L. Lovász, Perfect graphs, in: More selected topics in graph theory (L. W. Beineke and R. J. Wilson, eds.), to appear.

    Google Scholar 

  121. L. Lovász, Ear-decompositions of matching-covered graphs, preprint, 1982.

    Google Scholar 

  122. A. Lubiw, T-free matrices, M. Sc. thesis, Univ. of Waterloo, Waterloo, Ont., 1982.

    Google Scholar 

  123. C. L. Lucchesi, A minimax equality for directed graphs, Ph. D. thesis, Univ. of Waterloo, Waterloo, Ont., 1976.

    Google Scholar 

  124. C. L. Lucchesi and D. H. Younger, A minimax relation for directed graphs, J. London Math. Soc. (2) 17 (1978) 369–374.

    Article  MathSciNet  MATH  Google Scholar 

  125. W. Mader, Uber die Maximalzahl kantendisjunkter A-Wege, Arch. Math. (Basel) 30 (1978) 325–336.

    MathSciNet  MATH  Google Scholar 

  126. W. Mader, Uber die Maximalzahl kreuzungsfreier H-Wege, Arch. Math. (Basel) 31 (1978) 387–402.

    MathSciNet  Google Scholar 

  127. A. B. Marsh, Matching algorithms, Ph. D. thesis, Johns Hopkins Univ., Baltimore, 1979.

    Google Scholar 

  128. C. J. H. McDiarmid, The solution of a time-tabling problem, J. Inst. Maths. Appl. 9 (1972) 23–34.

    Article  MathSciNet  MATH  Google Scholar 

  129. K. Menger, Zur allgemeinen Kurventheorie, Fund. Math. 10 (1927) 96–115.

    MATH  Google Scholar 

  130. H. Meyniel, On the perfect graph conjecture, Discrete Math. 16 (1976) 339–342.

    Article  MathSciNet  Google Scholar 

  131. G. J. Minty, Monotone networks, Proc. Roy. Soc. London Ser. A 257 (1960) 194–212.

    Article  MathSciNet  MATH  Google Scholar 

  132. G. J. Minty, On maximal independent sets of vertices in a claw-free graph, J. Combinatorial Theory (B) 28 (1980) 284–304.

    Article  MathSciNet  MATH  Google Scholar 

  133. L. Mirsky, Transversal theory, Acad. Press, London, 1971.

    MATH  Google Scholar 

  134. C. St. J. A. Nash-Williams, Edge-disjoint spanning trees of finite graphs, J. London Math. Soc. 36 (1961) 445–450.

    Article  MathSciNet  MATH  Google Scholar 

  135. C. St. J. A. Nash-Williams, Decomposition of finite graphs into forests, J. London Math. Soc. 39 (1964) 12.

    Article  MathSciNet  MATH  Google Scholar 

  136. 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. H. Okamura and P. D. Seymour, Multicommodity flows in planar graphs, J. Combinatorial Theory (B) 31 (1981) 75–81.

    Article  MathSciNet  MATH  Google Scholar 

  138. A. Orden, The transshipment problem, Manag. Sci. 2 (1956) 276–285.

    MathSciNet  MATH  Google Scholar 

  139. 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. M. W. Padberg and M. R. Rao, Odd minimum cut-sets and b-matchings, Math, of Oper. Res. 7 (1982) 67–80.

    MathSciNet  MATH  Google Scholar 

  141. C. H. Papadimitriou and K. Steiglitz, Combinatorial optimization: algorithms and complexity, Prentice-Hall, Englewood Cliffs, N.J., 1982.

    MATH  Google Scholar 

  142. 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. H. Perfect, Applications of Menger’s graph theorem, J. Math. Analysis Appl. 22 (1968) 96–111.

    Article  MathSciNet  MATH  Google Scholar 

  144. W. R. Pulleyblank, Faces of matching polyhedra, Ph. D. thesis, Univ. of Waterloo, Waterloo, Ont., 1973.

    Google Scholar 

  145. W. R. Pulleyblank, Dual integrality in b-matching problems, Math. Programming Study 12 (1980) 176–196.

    MathSciNet  MATH  Google Scholar 

  146. W. R. Pulleyblank, Polyhedral combinatorics, this volume.

    Google Scholar 

  147. R. Rado, A note on independence functions, Proc. London Math. Soc. 7 (1957) 300–320.

    MathSciNet  MATH  Google Scholar 

  148. B. Rothschild and A. Whinston, On two-commodity network flows, Operations Res. 14 (1966) 377–387.

    Article  MathSciNet  MATH  Google Scholar 

  149. B. Rothschild and A. Whinston, Feasibility of two-commodity network flows, Operations Res. 14 (1966) 1121–1129.

    Article  MathSciNet  MATH  Google Scholar 

  150. N. Sbihi, Étude des stables dans les graphes sans étoile, M. Sc. thesis, Univ. Sci. et Méd. Grenoble, 1978.

    Google Scholar 

  151. N. Sbihi and J. P. Uhry, A class of h-perfect graphs, Rapport de Rech. No. 236, IRMA, Grenoble, 1981.

    Google Scholar 

  152. A. Schrijver, A counterexemple to a conjecture of Edmonds and Giles, Discrete Math. 32 (1980) 213–214.

    MathSciNet  MATH  Google Scholar 

  153. 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. a] A. Schrijver, Min-max relations for directed graphs, Annals of Discrete Math. 16 (1982) 261–280.

    MathSciNet  MATH  Google Scholar 

  155. 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. A. Schrijver, Total dual integrality from directed graphs, crossing families, and sub- and supermodular functions, Proc. Waterloo 1982, to appear.

    Google Scholar 

  157. a] A. Schrijver, Packing and covering of crossing families of cuts, Report AE 1/ 83, Univ. van Amsterdam, Amsterdam, 1983. 1983

    Google Scholar 

  158. b] A. Schrijver, Supermodular colourings, Report AE 4/83, Univ. van Amsterdam, Amsterdam, 1983.

    Google Scholar 

  159. 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. A. Schrijver and P. D. Seymour, Solution of two fractional packing problems of Lovasz, Discrete Math. 26 (1979) 177–184.

    Article  MathSciNet  MATH  Google Scholar 

  161. P. D. Seymour, The matroids with the max-flow min-cut property, J. Combina¬torial Theory (B) 23 (1977) 189–222.

    Article  MathSciNet  MATH  Google Scholar 

  162. a] P. D. Seymour, A two-commodity cut theorem, Discrete Math. 23 (1978) 177–181.

    Article  MathSciNet  MATH  Google Scholar 

  163. 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. P. D. Seymour, On multi-colourings of cubic graphs, and conjectures of Fulkerson and Tutte, Proc. London Math. Soc. (3) 38 (1979) 423–460.

    Article  MathSciNet  MATH  Google Scholar 

  165. b] P. D. Seymour, A short proof of the two-commodity flow theorem, J. Combinatorial Theory (B) 26 (1979) 370–371.

    Article  MathSciNet  MATH  Google Scholar 

  166. P. D. Seymour, Four-terminus flows, Networks 10 (1980) 79–86.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  168. P. D. Seymour, Matroids and multicommodity flows, Europ. J. Comb. 2 (1981) 257–290.

    MathSciNet  MATH  Google Scholar 

  169. F. Sterboul, A characterization of the graphs in which the transversal number equals the matching number, J. Combinatorial Theory (B) 27 (1979) 228–229.

    Article  MathSciNet  MATH  Google Scholar 

  170. J. Stoer and C. Witzgall, Convexity and optimization in finite dimensions I

    Google Scholar 

  171. Springer, Berlin, 1970.

    Google Scholar 

  172. W. T. Tutte, The factorization of linear graphs, J. London Math. Soc. 22 (1947) 107–111.

    Article  MathSciNet  MATH  Google Scholar 

  173. W. T. Tutte, The factors of graphs, Canad. J. Math. 4 (1952) 314–328.

    MathSciNet  MATH  Google Scholar 

  174. W. T. Tutte, The 1-factors of oriented graphs, Proc. Amer. Math. Soc. 4 (1953) 922–931.

    MathSciNet  MATH  Google Scholar 

  175. W. T. Tutte, A short proof of the factor theorem for finite graphs, Canad. J. Math. 6 (1954) 347–352.

    MathSciNet  MATH  Google Scholar 

  176. W. T. Tutte, On the problem of decomposing a graph into n connected factors, J. London Math. Soc. 36 (1961) 221–230.

    Article  MathSciNet  MATH  Google Scholar 

  177. W. T. Tutte, Graph factors, Combinatorica 1 (1981) 79–97.

    MathSciNet  MATH  Google Scholar 

  178. V. G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz. 3 (1964) 25–30 (in Russian).

    MathSciNet  Google Scholar 

  179. B. L. van der Waerden, Moderne Algebra, Springer, Berlin, 1937.

    Google Scholar 

  180. D. J. A. Welsh, On matroid theorems of Edmonds and Rado, J. London Math. Soc. 2 (1970) 251–256.

    Article  MathSciNet  MATH  Google Scholar 

  181. D. J. A. Welsh, Matroid theory, Acad. Press, London, 1976.

    MATH  Google Scholar 

  182. D. de Werra, On some combinatorial problems arising in scheduling, Canad. Oper. Res. Soc. J. 8 (1970) 165–175.

    Google Scholar 

  183. D. de Werra, Decomposition of bipartite multigraphs into matchings, Zeitschr. Oper. Res. 16 (1972) 85–90.

    Article  Google Scholar 

  184. H. Whitney, On the abstract properties of linear independence, Amer. J. Math. 57 (1935) 509–533.

    Article  MathSciNet  Google Scholar 

  185. R. J. Wilson, Introduction to graph theory, Oliver and Boyd, Edinburgh, 1972.

    MATH  Google Scholar 

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  1. Instituut voor Actuariaat en Econometrie, Universiteit van Amsterdam, Jodenbreestraat 23, 1011 NH, Amsterdam, The Netherlands

    A. Schrijver

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  1. Institut für Ökonometrie und Operations Research, Universität Bonn, Nassestraße 2, D-5300, Bonn 1, Germany

    Achim Bachem  & Bernhard Korte  & 

  2. Lehrstuhl für Angewandte Mathematik II, Universität Augsburg, Memminger Straße 6, D-8900, Augsburg, Germany

    Martin Grötschel

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Schrijver, A. (1983). Min-Max Results in Combinatorial Optimization. In: Bachem, A., Korte, B., Grötschel, M. (eds) Mathematical Programming The State of the Art. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-68874-4_18

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