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Dual Integrality in Combinatorial Optimization

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Handbook of Combinatorial Optimization

Abstract

The notion of total dual integrality and its variants are powerful tools to derive combinatorial min-max relation efficiently, yielding many fundamental results in combinatorial optimization. Following the pioneering work of Edmonds and Geiles (A min–max relation for submodular functions on graphs, in Annals of Discrete Mathematics, vol. 1, North-Holland, Amsterdam, 1977, pp. 185–204), considerable research efforts have been devoted to the study of the dual integrality from theoretical and algorithmic points of view. This chapter reviews significant progresses that have been made since the publication of Schrijver’s celebrated monograph (Combinatorial Optimization - Polyhedra and Efficiency, Springer, Berlin, 2003), which contained a comprehensive and in-depth treatment of the state-of-the-art in dual integrality theory and its application, among many others.

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Recommended Reading

  1. N. Apollonio, Integrality properties of edge path tree families. Discrete Math. 309, 4181–4184 (2009)

    MathSciNet  MATH  Google Scholar 

  2. M.L. Balinski, Integer programming: methods, uses, computation. Manag. Sci. Ser. A 12, 253–313 (1965)

    MathSciNet  MATH  Google Scholar 

  3. R. Bar-Yehuda, K. Bendel, A. Freund, D. Rawitz, Local ratio: A unified framework for approximation algorithms. ACM Comput. Surv. 36, 422–463 (2004)

    Google Scholar 

  4. C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1976)

    MATH  Google Scholar 

  5. S. Bessy, S. Thomassé, Spanning a strong digraph by \(\alpha\) circuits: A proof of Gallai’s conjecture. Combinatorica 27, 659–667 (2007)

    MathSciNet  MATH  Google Scholar 

  6. J.A. Bondy, U.S.R. Murty, Graph Theory with Applications (Macmillan, London, 1976)

    MATH  Google Scholar 

  7. F. Bonomo, G. Durán, M.C. Lin, J.L. Szwarcfiter, On balanced graphs. Math. Program. 105, 233–250 (2006)

    MathSciNet  MATH  Google Scholar 

  8. F. Bonomo, M. Chudnovsky, G. Durán, Partial characterizations of clique-perfect graphs I: subclasses of claw-free graphs. Discrete Appl. Math. 156, 1058–1082 (2008)

    MathSciNet  MATH  Google Scholar 

  9. F. Bonomo, M. Chudnovsky, G. Durán, Partial characterizations of clique-perfect graphs II: diamond-free and Helly circular-arc graphs. Discrete Math. 309, 3485–3499 (2009)

    MathSciNet  MATH  Google Scholar 

  10. F. Bonomo, G. Durán, M.C. M.D. Safe, A.K. Wagler, On minimal forbidden subgraph characterizations of balanced graphs. Electron. Note Discrete Math. 35, 41–46 (2009)

    Google Scholar 

  11. F. Bonomo, G. Durán, F. Soulignac, G. Sueiro, Partial characterizations of coordinated graphs: Line graphs and complements of forests. Math. Method Oper. Res. 69, 251–270 (2009)

    MATH  Google Scholar 

  12. E. Boros, K. Elbassioni, V. Gurvich, H.R. Tiwary, The negative cycles polyhedron and hardness of checking some polyhedral properties. Ann. Oper. Res. 188, 63–76 (2011)

    MathSciNet  MATH  Google Scholar 

  13. M. Cai, X. Deng, W. Zang, An approximation algorithm for feedback vertex sets in tournaments. SIAM J. Comput. 30, 1993–2007 (2001)

    MathSciNet  MATH  Google Scholar 

  14. M. Cai, X. Deng, W. Zang, A min–max theorem on feedback vertex sets. Math. Oper. Res. 27, 361–371 (2002)

    MathSciNet  MATH  Google Scholar 

  15. P. Charbit, A. Sebö, Cyclic orders: Equivalence and duality. Combinatorica 28, 131–143 (2008)

    MathSciNet  MATH  Google Scholar 

  16. Q. Chen, X. Chen, Packing cycles exactly in polynomial time. J. Combin. Optim. 23, 167–188 (2012)

    MATH  Google Scholar 

  17. X. Chen, Z. Chen, W. Zang, A unified approach to box-Mengerian hypergraphs. Math. Oper. Res. 35, 655–668 (2010)

    MathSciNet  MATH  Google Scholar 

  18. X. Chen, G. Ding, X. Hu, W. Zang, A min–max relation on packing feedback vertex sets. Math. Oper. Res. 31, 777–788 (2006)

    MathSciNet  MATH  Google Scholar 

  19. X. Chen, G. Ding, W. Zang, The box-TDI system associated with 2-edge connected spanning subgraphs. Discrete Appl. Math. 157, 118–125 (2009)

    MathSciNet  MATH  Google Scholar 

  20. X. Chen, G. Ding, W. Zang, A characterization of box-Mengerian matroid ports. Math. Oper. Res. 33, 497–512 (2008)

    MathSciNet  MATH  Google Scholar 

  21. X. Chen, X. Hu, W. Zang, A min–max theorem on tournaments. SIAM J. Comput. 37, 923–937 (2007)

    MathSciNet  MATH  Google Scholar 

  22. M. Chudnovsky, G. Cornuéjols, X. Liu, P.D. Seymour, K. Vušković, Recognizing Berge graphs. Combinatorica 25, 143–186 (2005)

    MathSciNet  MATH  Google Scholar 

  23. M. Chudnovsky, N. Robertson, P.D. Seymour, R. Thomas, The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)

    MathSciNet  MATH  Google Scholar 

  24. V. Chvátal, On certain polytopes associated with graphs. J. Combin. Theor Ser. B 18, 138–154 (1975)

    MATH  Google Scholar 

  25. W. Cook, On box totally dual integral polyhedra. Math. Program. 34, 48–61 (1986)

    MATH  Google Scholar 

  26. W. Cook, L. Lovász, A. Schrijver, A polynomial-time test for total dual integrality in fixed dimension. Math. Program. Study 22, 64–69 (1984)

    MATH  Google Scholar 

  27. M. Conforti, G. Cornuéjols, K. Vušković, Balanced matrices. Discrete Math. 306, 2411–2437 (2006)

    MathSciNet  MATH  Google Scholar 

  28. G. Cornuéjols, Combinatorial Optimization: Packing and Covering (SIAM, Philadelphia, 2001)

    Google Scholar 

  29. G. Cornuéjols, J. Fonlupt, D. Naddef, The traveling salesman problem on a graph and somed related integer polyhedra. Math. Program. 33, 1–27 (1985)

    MATH  Google Scholar 

  30. G. Cornuéjols, B. Guenin, F. Margot, The packing property. Math. Program. 89, 113–126 (2000)

    MathSciNet  MATH  Google Scholar 

  31. G. Cornuéjols, D. Hartvigsen, An extension of matching theory. J. Combin. Theor Ser. B 40, 285–296 (1986)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  33. R. Diestel, Graph Theory, 3rd edn. (Springer, New York, 2005)

    MATH  Google Scholar 

  34. G. Ding, Clutters with \(\tau _{2} = 2\tau\). Discrete Math. 115, 141–152 (1993)

    MathSciNet  MATH  Google Scholar 

  35. G. Ding, L. Feng, W. Zang, The complexity of recognizing linear systems with certain integrality properties. Math. Program. Ser. A 114, 321–334 (2008)

    MathSciNet  MATH  Google Scholar 

  36. G. Ding, W. Zang, Packing cycles in graphs. J. Combin. Theor Ser. B 86, 381–407 (2002)

    MathSciNet  MATH  Google Scholar 

  37. G. Ding, W. Zang, Packing circuits in matroids. Math. Program. Ser. A 119, 137–168 (2009)

    MathSciNet  MATH  Google Scholar 

  38. J. Edmonds, Submodular functions, matroids, and certain polyhedra, in Combinatorial Structures and Their Applications (Proceedings Calgary International Conference on Combinatorial Structures and Their Applications, Calgary, Alberta, 1969) ed. by R. Guy, H. Hanani, N. Sauer, J. Schöonheim (Gordon and Breach, New York, 1970), pp. 69–87

    Google Scholar 

  39. J. Edmonds, Edge-disjoint branchings, in Combinatorial Algorithmis (Algorithmic Press, New York, 1973), pp. 91–96

    Google Scholar 

  40. J. Edmonds, D.R. Fulkerson, Bottleneck extrema. J. Combin. Theor 8, 299–306 (1970)

    MathSciNet  MATH  Google Scholar 

  41. J. Edmonds, R. Giles, A min–max relation for submodular functions on graphs, in Annals of Discrete Mathematics, vol.  1 (North-Holland, Amsterdam, 1977), pp. 185–204

    Google Scholar 

  42. J. Edmonds, R. Giles, Total dual integrality of linear inequality systems, in Progress in Combinatorial Optimization, ed. by W.R. Pulleyblank (Academic, Toronto, 1984), pp. 117–129

    Google Scholar 

  43. P. Erdös, L. Pósa, On the independent circuits contained in a graph. Can. J. Math. 17, 347–352 (1965)

    MATH  Google Scholar 

  44. U. Faigle, W. Kern, Submodular linear programs on forests. Math. Program. 72, 195–206 (1996)

    MathSciNet  MATH  Google Scholar 

  45. U. Faigle, W. Kern, On the core of ordered submodular cost games. Math. Program. Ser. A 87, 483–499 (2000)

    MathSciNet  MATH  Google Scholar 

  46. T. Feder, A new fixed point approach for stable networks and stable marriages. J. Comput. Syst. Sci. 45, 233–284 (1992)

    MathSciNet  MATH  Google Scholar 

  47. P. Feofiloff, D.H. Younger, Directed cut transversal packing for source-sink connected graphs. Combinatroica 7, 255–263 (1987)

    MathSciNet  MATH  Google Scholar 

  48. S. Fiorini, N. Hardy, Nadia, B. Reed, A. Vetta, Approximate min–max relations for odd cycles in planar graphs. Math. Program. 110, 71–91 (2007)

    MathSciNet  MATH  Google Scholar 

  49. J. Fonlupt, D. Naddef, The traveling salesman problem in graphs with some excluded minors. Math. Program. 53, 147–172 (1992)

    MathSciNet  MATH  Google Scholar 

  50. A. Frank, A coloring question on digraphs, in DMANET, 1998

    Google Scholar 

  51. A. Frank, Rooted k-connections in digraphs. Discrete Appl. Math. 157, 1242–1254 (2009)

    MathSciNet  MATH  Google Scholar 

  52. A. Frank, T. Király, A Survey on covering supermodular functions, in Research Trends in Combinatorial Optimization (Boon, 2008) ed. by W. Cook, L. Lovász, J. Vygen, (Springer, Berlin, 2009), pp. 87–126

    Google Scholar 

  53. A. Frank, T. Király, Z. Király, On the orientation of graphs and hypergraphs. Discrete Appl. Math. 131, 385–400 (2003)

    MathSciNet  MATH  Google Scholar 

  54. A. Frank, É. Tardos, An application of submodular flows. Lin. Algebra Appl. 114–115, 329–348 (1989)

    MathSciNet  Google Scholar 

  55. S. Fujishige, Submodular Functions and Optimization, 2nd edn. Annals of Discrete Mathematics, vol. 58 (Elsevier, London, 2005)

    MATH  Google Scholar 

  56. D.R. Fulkerson, Networks, frames, and blocking systems, in Mathematics of the Decision Sciences, Part I, ed. by G.B. Dantzig, A.F. Veinott (American Mathematical Society, Providence, Rhode Island, 1968), pp. 303–334

    Google Scholar 

  57. D.R. Fulkerson, Blocking and antiblocking pairs of polyhedra. Math. Program. 1, 168–194 (1971)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  59. D. Gale, L. Shapley, College admissions and the stability of marriage. Am. Math. Mon. 69, 9–15 (1962)

    MathSciNet  MATH  Google Scholar 

  60. M.R. Garey, D.S. Johnson, Computers and Intractability (W.H. Freeman and Company, New York, 1979)

    MATH  Google Scholar 

  61. J. Geelen, B. Guenin, B, Packing odd circuits in Eulerian graphs. J. Combin. Theor Ser. B 86, 280–295 (2002)

    MathSciNet  MATH  Google Scholar 

  62. A.M.H. Gerards, M. Laurent, A characterization of box \(\frac{1} {d}\)-integral binary clutters. J. Combin. Theor Ser. B 65, 186–207 (1995)

    MathSciNet  MATH  Google Scholar 

  63. M.X. Goemans, D.P. Williamson, The primal-dual method for approximation algorithms and its application to network design problems, in Approximation Algorithms for NP-Hard Problems, ed. by D.S. Hochbaum (PWS Publishing Company, Boston, MA), pp. 144–191

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  65. M. Grötschel, L. Lovász, A. Schrijver, Geometric Algorithms and Combinatorial Optimization (Springer, Berlin, 1988)

    MATH  Google Scholar 

  66. M. Grötschel, C. Monma, M. Stoer, Facets for polyhedra arising in the design of communication networks with low-connectivity constraints. SIAM J. Optim. 2, 474–504 (1992)

    MathSciNet  MATH  Google Scholar 

  67. M. Grötschel, C. Monma, M. Stoer, Computational results with a cutting plane algorithm for designing communication networks with low-connectivity constraints. Oper. Res. 40, 309–330 (1992)

    MathSciNet  MATH  Google Scholar 

  68. B. Guenin, A short proof of Seymour’s characterization of the matroids with the max-flow min-cut property. J. Combin. Theor Ser. B 86, 273–279 (2002)

    MathSciNet  MATH  Google Scholar 

  69. B. Guenin, R. Thomas, Packing directed circuits exactly. Combinatorica 31, 397–421 (2011)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  71. H. Hirai, Tight spans of distances and the dual fractionality of undirected multiflow problems. J. Combin. Theor Ser. B 99, 843–868 (2009)

    MathSciNet  MATH  Google Scholar 

  72. T.C. Hu, Multi-commodity network flows. Oper. Res. 11, 344–360 (1963)

    MATH  Google Scholar 

  73. K. Ken-ichi, Half integral packing, Erdös-Posá-property and graph minors, in Proceedings of the 18th annual ACM-SIAM Symposium on Discrete Algorithms, 2007, pp. 1187–1196

    Google Scholar 

  74. K. Ken-ichi, R. Bruce, A nearly linear time algorithm for the half integral disjoint paths packing, in Proceedings of the 19th annual ACM-SIAM Symposium on Discrete Algorithms, 2008, pp. 446–454

    Google Scholar 

  75. S. Khanna, J. Naor, F.B. Shepherd, Directed network design with orientation constraints. SIAM J. Discrete Math. 19, 245–257 (2005)

    MathSciNet  MATH  Google Scholar 

  76. T. Király, J. Pap, Total dual integrality of Rothblum’s description of the stable-marriage polyhedron. Math. Oper. Res. 33, 283–290 (2008)

    MathSciNet  MATH  Google Scholar 

  77. D. Král, H. Voss, Edge-disjoint odd cycles in planar graphs. J. Combin. Theor Ser. B 90, 107–120 (2004)

    MATH  Google Scholar 

  78. M. Laurent, S. Poljak, One-third-integrality in the max-cut problem. Math. Program. 71, 29–50 (1995)

    MathSciNet  MATH  Google Scholar 

  79. O. Lee, Y. Wakabayashi, Note on a min–max conjecture of Woodall. J. Graph Theorem 38, 36–41 (2001)

    MathSciNet  MATH  Google Scholar 

  80. A. Lehman, On the length-width inequality. Math. Program. 17, 403–417 (1979)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  83. A.R. Mahjoub, Two-edge connected spanning subgraphs and polyhedra. Math. Program. 64, 199–208 (1994)

    MathSciNet  MATH  Google Scholar 

  84. A.R. Mahjoub, On perfectly two-edge connected graphs. Discrete Math. 170, 153–172 (1997)

    MathSciNet  MATH  Google Scholar 

  85. C.L. Monma, B.S. Munson, W.R. Pulleyblank, Minimum weight 2-connected spanning networks. Math. Program. 46, 153–172 (1990)

    MathSciNet  MATH  Google Scholar 

  86. J. von Neumann, O. Morgenstern, Theory of Games and Economic Behavior (Princeton University Press, Princeton, 1944)

    MATH  Google Scholar 

  87. E. O’Shea, A. Sebö, Alternatives for testing total dual integrality. Math. Program. 132, 57–78 (2012)

    MathSciNet  MATH  Google Scholar 

  88. J. Oxley, Matroid Theory (Oxford University Press, Oxford, 1992)

    MATH  Google Scholar 

  89. J. Pap, Recognizing conic TDI systems is hard. Math. Program. Ser. A 128, 43–48 (2011)

    MathSciNet  MATH  Google Scholar 

  90. G. Pap, Weighted restricted 2-matching. Math. Program. Ser. A 119, 305–329 (2009)

    MathSciNet  MATH  Google Scholar 

  91. A.D. Pia, G. Zambelli, Half-integral vertex covers on bipartite bidirected braphs: total dual integrality and cut-rank. SIAM J. Discrete Math. 23, 1281–1296 (2009)

    MathSciNet  MATH  Google Scholar 

  92. W.R. Pulleyblank, Polyhedral combinatorics, in Mathematical Programming – The State of the Art (Bonn, 1982), ed. by A. Bachem, M. Grötschel, B. Korte (Springer, Berlin, 1983), pp. 312–345

    Google Scholar 

  93. W. Pulleyblank, J. Edmonds, Facets of 1-matching polyhedra, in Hypergraph Seminar (Proceedings Working Seminar on Hypergraphs, Columbus, Ohio, 1972), ed. by C. Berge, D. Ray-Chaudhuri (Springer, Berlin, 1974), pp. 214–242

    Google Scholar 

  94. R. Rado, Bemerkungen zur Kombinatorik im AnschluSS an Untersuchungen von Herrn D. König. Sitzungsberichte der Berliner Mathematischen Gesellschaft 32, 60–75 (1933)

    Google Scholar 

  95. U. Rothblum, Characterization of stable matchings as extreme points of a polytope. Math. Program. 54, 57–67 (1992)

    MathSciNet  MATH  Google Scholar 

  96. T.J. Schaefer, The complexity of satisfiability problems, in Proceedings of 10th ACM Symposimum on Theory of Computing, New York, 1986, pp. 216–226

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  98. A. Schrijver, Min-max relations for directed graphs. Ann. Discrete Math. 16, 127–146 (1982)

    MathSciNet  Google Scholar 

  99. A. Schrijver, Min-max results in combinatorial optimization, in Mathematical Programming: The State of the Art Bonn 1982, ed. by A. Bachem, M. Grötschel, B. Korte (Springer, New York, 1983), pp. 439–500

    Google Scholar 

  100. A. Schrijver, Total dual integrality from directed graphs, crossing families, and sub- and supermodular functions, in Progress in Combinatorial Optimization (Academic Press, Toronto, 1984), pp. 315–361

    Google Scholar 

  101. A. Schrijver, Theory of Linear and Integer Programming (Wiley, New York, 1986)

    MATH  Google Scholar 

  102. A. Schrijver, Polyhedral combinatorics, in Handbook of Combinatorics, vol. 2, ed. by R.L. Graham, M. Grötschel, L. Lovász (Elsevier, Amsterdam, 1995), pp. 1649–1704

    Google Scholar 

  103. A. Schrijver, Combinatorial Optimization - Polyhedra and Efficiency (Springer, Berlin, 2003)

    MATH  Google Scholar 

  104. A. Schrijver, P.D. Seymour, A proof of total dual integrality of matching polyhedra, in Mathematical Centre Report ZN 79/77 (Mathematical Centre, Amsterdam, 1977)

    Google Scholar 

  105. A. Sebö, Minmax relations for cyclically ordered digraphs. J. Combin. Theor Ser. B 97, 518–552 (2007)

    MATH  Google Scholar 

  106. P.D. Seymour, The forbidden minors of binary clutters. J. Lond. Math. Soc. 12, 356–360 (1976)

    MathSciNet  MATH  Google Scholar 

  107. P.D. Seymour, The matroids with the max-flow min-cut property. J. Combin. Theor Ser. B 23, 189–222 (1977)

    MathSciNet  MATH  Google Scholar 

  108. P.D. Seymour, Decomposition of regular matroids. J. Combin. Theor Ser. B 28, 305–359 (1980)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  110. E. Speckenmeyer, On feedback problems in digraphs, in Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, vol. 411 (Springer, Berlin, 1989), pp. 218–231

    Google Scholar 

  111. K. Truemper, Matroid Decomposition (Academic Press, Toronto, 1992)

    MATH  Google Scholar 

  112. F.T. Tseng, K. Truemper, A decomposition of the matroids with the max-flow min-cut property. Discrete Appl. Math. 15, 329–364 (1986)

    MathSciNet  MATH  Google Scholar 

  113. D. Vandenbussche, G. Nemhauser, The 2-edge-connected subgraph polyhedron. J. Combin. Optim. 9, 357–379 (2005)

    MathSciNet  MATH  Google Scholar 

  114. D.R. Woodall, Menger and König systems, in Theory and Applications of Graphs, Lecture Notes in Mathematics, vol. 642 (1978), pp. 620–635

    Google Scholar 

  115. M. Yuji, Fractional packing in ideal clutters, in Proceedings of the 18th annual ACM-SIAM Symposium on Discrete Algorithms, 2007, pp. 1181–1186

    Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing, People’s Republic of China

    Xujin Chen

  2. Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing, People’s Republic of China

    Xiaodong Hu

  3. Department of Mathematics, The University of Hong Kong, Hong Kong, Hong Kong

    Wenan Zang

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  1. Xujin Chen
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  2. Xiaodong Hu
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  3. Wenan Zang
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Correspondence to Xujin Chen .

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Editors and Affiliations

  1. Department of Industrial and Systems Eng, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Department of Computer Science, University of Texas, Dallas, Richardson, Texas, USA

    Ding-Zhu Du

  3. Dept. Comp. Sci. & Engineering, University of California, San Diego, La Jolla, California, USA

    Ronald L. Graham

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Chen, X., Hu, X., Zang, W. (2013). Dual Integrality in Combinatorial Optimization. In: Pardalos, P., Du, DZ., Graham, R. (eds) Handbook of Combinatorial Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7997-1_57

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