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Integer Programming pp 351–388Cite as

Enumeration

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Part of the book series: Graduate Texts in Mathematics ((GTM,volume 271))

Abstract

The goal of this chapter is threefold. First we present a polynomial algorithm for integer programming in fixed dimension. This algorithm is based on elegant ideas such as basis reduction and the flatness theorem. Second we revisit branch-and-cut, the most successful approach in practice for a wide range of applications. In particular we address a number of implementation issues related to the enumerative aspects of branch-and-cut. Finally we present an approach for dealing with integer programs that have a high degree of symmetry.

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Bibliography

  1. K. Aardal, R.E. Bixby, C.A.J. Hurkens, A.K. Lenstra, J.W. Smeltink, Market split and basis reduction: towards a solution of the Cornuéjols–Dawande instances. INFORMS J. Comput. 12, 192–202 (2000)

    MATH  MathSciNet  Google Scholar 

  2. K. Aardal, A.K. Lenstra, Hard equality constrained integer knapsacks. Math. Oper. Res. 29, 724–738 (2004); Erratum: Math. Oper. Res. 31, 846 (2006)

    Google Scholar 

  3. K. Aardal, C. Hurkens, A.K. Lenstra, Solving a system of diophantine equations with lower and upper bounds on the variables. Math. Oper. Res. 25, 427–442 (2000)

    MATH  MathSciNet  Google Scholar 

  4. K. Aardal, R. Weismantel, L.A. Wolsey, Non-standard approaches to integer programming. Discrete Appl. Math. 123, 5–74 (2002)

    MATH  MathSciNet  Google Scholar 

  5. T. Achterberg, Constraint Integer Programming. Ph.D. thesis, ZIB, Berlin, 2007

    Google Scholar 

  6. T. Achterberg, T. Berthold, Improving the feasibility pump. Discrete Optim. 4, 77–86 (2007)

    MATH  MathSciNet  Google Scholar 

  7. T. Achterberg, T. Koch, A. Martin, Branching rules revisited. Oper. Res. Lett. 33, 42–54 (2005)

    MATH  MathSciNet  Google Scholar 

  8. T. Achterberg, T. Koch, A. Martin, MIPLIB 2003. Oper. Res. Lett. 34, 361–372 (2006)

    MATH  MathSciNet  Google Scholar 

  9. M. Ajtai, The shortest vector problem in L2 is NP-hard for randomized reductions, in Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC-98), (1998), pp. 10–19

    Google Scholar 

  10. F. Alizadeh, Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim. 5, 13–51 (1995)

    MATH  MathSciNet  Google Scholar 

  11. K. Andersen, G. Cornuéjols, Y. Li, Split closure and intersection cuts. Math. Program. A 102, 457–493 (2005)

    MATH  Google Scholar 

  12. K. Andersen, Q. Louveaux, R. Weismantel, L.A. Wolsey, Inequalities from two rows of a simplex tableau, in Proceedings of IPCO XII, Ithaca, NY. Lecture Notes in Computer Science, vol. 4513 (2007), pp. 1–15

    MathSciNet  Google Scholar 

  13. D. Applegate, R.E. Bixby, V. Chvátal, W.J. Cook, The Traveling Salesman Problem. A Computational Study (Princeton University Press, Princeton, 2006)

    Google Scholar 

  14. S. Arora, B. Barak, Complexity Theory: A Modern Approach (Cambridge University Press, Cambridge, 2009)

    Google Scholar 

  15. A. Atamtürk, Strong formulations of robust mixed 0–1 programming. Math. Program. 108, 235–250 (2006)

    MATH  MathSciNet  Google Scholar 

  16. A. Atamtürk, G.L. Nemhauser, M.W.P. Savelsbergh, Conflict graphs in solving integer programming problems. Eur. J. Oper. Res. 121, 40–55 (2000)

    MATH  Google Scholar 

  17. R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows (Prentice Hall, New Jersey, 1993)

    MATH  Google Scholar 

  18. G. Averkov, On maximal S-free sets and the Helly number for the family of S-convex sets. SIAM J. Discrete Math. 27(3), 1610–1624 (2013)

    MATH  MathSciNet  Google Scholar 

  19. G. Averkov, A. Basu, On the unique lifting property, IPCO 2014, Bonn, Germany, Lecture Notes in Computer Science, 8494, 76–87 (2014)

    Google Scholar 

  20. D. Avis, K. Fukuda, A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete Comput. Geom. 8, 295–313 (1992)

    MATH  MathSciNet  Google Scholar 

  21. A. Bachem, R. von Randow, Integer theorems of Farkas lemma type, in Operations Research Verfahren/ Methods of Operations Research 32, III Symposium on Operations Research, Mannheim 1978, ed. by W. Oettli, F. Steffens (Athenäum, Königstein, 1979), pp. 19–28

    Google Scholar 

  22. E. Balas, Intersection cuts—a new type of cutting planes for integer programming. Oper. Res. 19, 19–39 (1971)

    MATH  MathSciNet  Google Scholar 

  23. E. Balas, Integer programming and convex analysis: intersection cuts from outer polars. Math. Program. 2 330–382 (1972)

    MATH  MathSciNet  Google Scholar 

  24. E. Balas, Disjunctive programming: properties of the convex hull of feasible points, GSIA Management Science Research Report MSRR 348, Carnegie Mellon University (1974); Published as invited paper in Discrete Appl. Math. 89, 1–44 (1998)

    Google Scholar 

  25. E. Balas, Facets of the knapsack polytope. Math. Program. 8, 146–164 (1975)

    MATH  MathSciNet  Google Scholar 

  26. E. Balas, Disjunctive programming and a hierarchy of relaxations for discrete optimization problems. SIAM J. Algebr. Discrete Methods 6, 466–486 (1985)

    MATH  MathSciNet  Google Scholar 

  27. E. Balas, A modified lift-and-project procedure. Math. Program. 79, 19–31 (1997)

    MATH  MathSciNet  Google Scholar 

  28. E. Balas, P. Bonami, Generating lift-and-project cuts from the LP simplex tableau: open source implementation and testing of new variants. Math. Program. Comput. 1, 165–199 (2009)

    MATH  MathSciNet  Google Scholar 

  29. E. Balas, S. Ceria, G. Cornuéjols, A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58, 295–324 (1993)

    MATH  Google Scholar 

  30. E. Balas, S. Ceria, G. Cornuéjols, R.N. Natraj, Gomory cuts revisited. Oper. Res. Lett. 19, 1–9 (1996)

    MATH  MathSciNet  Google Scholar 

  31. E. Balas, R. Jeroslow, Strengthening cuts for mixed integer programs. Eur. J. Oper. Res. 4, 224–234 (1980)

    MATH  MathSciNet  Google Scholar 

  32. E. Balas, M. Perregaard, A precise correspondence between lift-and-project cuts, simple disjunctive cuts and mixed integer Gomory cuts for 0–1 programming. Math. Program. B 94, 221–245 (2003)

    MATH  MathSciNet  Google Scholar 

  33. E. Balas, W.R. Pulleyblank, The perfectly matchable subgraph polytope of an arbitrary graph. Combinatorica 9, 321–337 (1989)

    MATH  MathSciNet  Google Scholar 

  34. E. Balas, A. Saxena, Optimizing over the split closure. Math. Program. 113, 219–240 (2008)

    MATH  MathSciNet  Google Scholar 

  35. W. Banaszczyk, A.E. Litvak, A. Pajor, S.J. Szarek, The flatness theorem for nonsymmetric convex bodies via the local theory of Banach spaces. Math. Oper. Res. 24 728–750 (1999)

    MATH  MathSciNet  Google Scholar 

  36. F. Barahona, R. Anbil, The volume algorithm: producing primal solutions with a subgradient method. Math. Program. 87, 385–399 (2000)

    MATH  MathSciNet  Google Scholar 

  37. I. Barany, T.J. Van Roy, L.A. Wolsey, Uncapacitated lot-sizing: the convex hull of solutions. Math. Program. 22, 32–43 (1984)

    MATH  Google Scholar 

  38. A. Barvinok, A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. Math. Oper. Res. 19, 769–779 (1994)

    MATH  MathSciNet  Google Scholar 

  39. A. Barvinok, A Course in Convexity. Graduate Studies in Mathematics, vol. 54 (American Mathematical Society, Providence, 2002)

    Google Scholar 

  40. A. Basu, M. Campelo, M. Conforti, G. Cornuéjols, G. Zambelli, On lifting integer variables in minimal inequalities. Math. Program. A 141, 561–576 (2013)

    MATH  Google Scholar 

  41. A. Basu, M. Conforti, G. Cornuéjols, G. Zambelli, Maximal lattice-free convex sets in linear subspaces. Math. Oper. Res. 35, 704–720 (2010)

    MATH  MathSciNet  Google Scholar 

  42. A. Basu, M. Conforti, G. Cornuéjols, G. Zambelli, Minimal inequalities for an infinite relaxation of integer programs. SIAM J. Discrete Math. 24, 158–168 (2010)

    MATH  MathSciNet  Google Scholar 

  43. A. Basu, R. Hildebrand, M. Köppe, M. Molinaro, A (k+1)-Slope Theorem for the k-Dimensional Infinite Group Relaxation. SIAM J. Optim. 23(2), 1021–1040 (2013)

    MATH  MathSciNet  Google Scholar 

  44. A. Basu, R. Hildebrand, M. Köppe, Equivariant perturbation in Gomory and Johnson infinite group problem III. Foundations for the k-dimensional case with applications to the case k = 2. www.optimization-online.org (2014)

  45. D.E. Bell, A theorem concerning the integer lattice. Stud. Appl. Math. 56, 187–188 (1977)

    MATH  Google Scholar 

  46. R. Bellman, Dynamic Programming (Princeton University Press, Princeton, 1957)

    MATH  Google Scholar 

  47. J.F. Benders, Partitioning procedures for solving mixed variables programming problems. Numerische Mathematik 4, 238–252 (1962)

    MATH  MathSciNet  Google Scholar 

  48. M. Bénichou, J.M. Gauthier, P. Girodet, G. Hentges, G. Ribière, O. Vincent, Experiments in mixed-integer linear programming. Math. Program. 1, 76–94 (1971)

    MATH  Google Scholar 

  49. A. Ben-Tal, A.S. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. MPS/SIAM Series in Optimization (SIAM, Philadelphia, 2001)

    Google Scholar 

  50. C. Berge, Two theorems in graph theory. Proc. Natl. Acad. Sci. USA 43, 842–844 (1957)

    MATH  MathSciNet  Google Scholar 

  51. D. Bertsimas, R. Weismantel, Optimization over Integers (Dynamic Ideas, Belmont, 2005)

    Google Scholar 

  52. D. Bienstock, M. Zuckerberg, Subset algebra lift operators for 0–1 integer programming. SIAM J. Optim. 15, 63–95 (2004)

    MATH  MathSciNet  Google Scholar 

  53. L.J. Billera, A. Sarangarajan, All 0,1 polytopes are traveling salesman polytopes. Combinatorica 16, 175–188 (1996)

    MATH  MathSciNet  Google Scholar 

  54. S. Binato, M.V.F. Pereira, S. Granville, A new Benders decomposition approach to solve power transmission network design problems. IEEE Trans. Power Syst. 16, 235–240 (2001)

    Google Scholar 

  55. J. R. Birge, F. Louveaux, Introduction to Stochastic Programming (Springer, New York, 2011)

    MATH  Google Scholar 

  56. R.E. Bixby, S. Ceria, C.M. McZeal, M.W.P. Savelsbergh, An updated mixed integer programming library: MIPLIB 3.0. Optima 58, 12–15 (1998)

    Google Scholar 

  57. R.E. Bixby, M. Fenelon, Z. Gu, E. Rothberg, R. Wunderling, Mixed integer programming: a progress report, in The Sharpest Cut: The Impact of Manfred Padberg and His Work, ed. by M. Grötschel. MPS/SIAM Series in Optimization (2004), pp. 309–326

    Google Scholar 

  58. P. Bonami, On optimizing over lift-and-project closures. Math. Program. Comput. 4, 151–179 (2012)

    MATH  MathSciNet  Google Scholar 

  59. P. Bonami, M. Conforti, G. Cornuéjols, M. Molinaro, G. Zambelli, Cutting planes from two-term disjunctions. Oper. Res. Lett. 41, 442–444 (2013)

    MATH  MathSciNet  Google Scholar 

  60. P. Bonami, G. Cornuéjols, S. Dash, M. Fischetti, A. Lodi, Projected Chvátal-Gomory cuts for mixed integer linear programs. Math. Program. 113, 241–257 (2008)

    MATH  MathSciNet  Google Scholar 

  61. P. Bonami, F. Margot, Cut generation through binarization, IPCO 2014, eds. by J. Lee, J. Vygen. LNCS, vol 8494 (2014) pp. 174–185

    Google Scholar 

  62. J.A. Bondy, U.S.R. Murty, Graph Theory (Springer, New York, 2008)

    MATH  Google Scholar 

  63. V. Borozan, G. Cornuéjols, Minimal valid inequalities for integer constraints. Math. Oper. Res. 34, 538–546 (2009)

    MATH  MathSciNet  Google Scholar 

  64. O. Briant, C. Lemaréchal, Ph. Meurdesoif, S. Michel, N. Perrot, F. Vanderbeck, Comparison of bundle and classical column generation. Math. Program. 113, 299–344 (2008)

    MATH  MathSciNet  Google Scholar 

  65. C.A. Brown, L. Finkelstein, P.W. Purdom, Backtrack Searching in the Presence of Symmetry, Nordic Journal of Computing 3, 203–219 (1996)

    MathSciNet  Google Scholar 

  66. S. Burer, D. Vandenbussche, Solving lift-and-project relaxations of binary integer programs. SIAM J. Optim. 16, 726–750 (2006)

    MATH  MathSciNet  Google Scholar 

  67. A. Caprara, M. Fischetti, \(\{0, \frac{1} {2}\}\) Chvátal–Gomory cuts. Math. Program. 74, 221–235 (1996)

    Google Scholar 

  68. A. Caprara, A.N. Letchford, On the separation of split cuts and related inequalities. Math. Program. B 94, 279–294 (2003)

    MATH  MathSciNet  Google Scholar 

  69. R.D. Carr, G. Konjevod, G. Little, V. Natarajan, O. Parekh, Compacting cuts: new linear formulation for minimum cut. ACM Trans. Algorithms 5, 27:1–27:6 (2009)

    Google Scholar 

  70. E. Chlamtac, M. Tulsiani, Convex relaxations and integrality gaps, in Handbook on Semidefinite, Conic and Polynomial Optimization, International Series in Operations Research and Management Science, Springer, vol. 166 (2012), pp. 139–169

    MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  73. V. Chvátal, Edmonds polytopes and a hierarchy of combinatorial optimization. Discrete Math. 4, 305–337 (1973)

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  75. V. Chvátal, W. Cook, M. Hartmann, On cutting-plane proofs in combinatorial optimization. Linear Algebra Appl. 114/115, 455–499 (1989)

    Google Scholar 

  76. M. Conforti, G. Cornuéjols, A. Daniilidis, C. Lemaréchal, J. Malick, Cut-generating functions and S-free sets, Math. Oper. Res. http://dx.doi.org/10.1287/moor.2014.0670

  77. M. Conforti, G. Cornuéjols, G. Zambelli, A geometric perspective on lifting. Oper. Res. 59, 569–577 (2011)

    MATH  MathSciNet  Google Scholar 

  78. M. Conforti, G. Cornuéjols, G. Zambelli, Equivalence between intersection cuts and the corner polyhedron. Oper. Res. Lett. 38, 153–155 (2010)

    MATH  MathSciNet  Google Scholar 

  79. M. Conforti, G. Cornuéjols, G. Zambelli, Extended formulations in combinatorial optimization. 4OR 8, 1–48 (2010)

    Google Scholar 

  80. M. Conforti, G. Cornuéjols, G. Zambelli, Corner polyhedron and intersection cuts. Surv. Oper. Res. Manag. Sci. 16, 105–120 (2011)

    Google Scholar 

  81. M. Conforti, M. Di Summa, F. Eisenbrand, L.A. Wolsey, Network formulations of mixed-integer programs. Math. Oper. Res. 34, 194–209 (2009)

    MATH  MathSciNet  Google Scholar 

  82. M. Conforti, L.A. Wolsey, Compact formulations as unions of polyhedra. Math. Program. 114, 277–289 (2008)

    MATH  MathSciNet  Google Scholar 

  83. M. Conforti, L.A. Wolsey, G. Zambelli, Split, MIR and Gomory inequalities (2012 submitted)

    Google Scholar 

  84. S.A. Cook, The complexity of theorem-proving procedures, in Proceedings 3rd STOC (Association for Computing Machinery, New York, 1971), pp. 151–158

    Google Scholar 

  85. W.J. Cook, Fifty-plus years of combinatorial integer programming, in 50 Years of Integer Programming 1958–2008, ed. by M. Jünger et al. (Springer, Berlin, 2010), pp. 387–430

    Google Scholar 

  86. W.J. Cook, In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation (Princeton University Press, Princeton, 2012)

    Google Scholar 

  87. W.J. Cook, W.H. Cunningham, W.R. Pulleyblank, A. Schrijver, Combinatorial Optimization (Wiley, New York, 1998)

    MATH  Google Scholar 

  88. W.J. Cook, S. Dash, R. Fukasawa, M. Goycoolea, Numerically accurate Gomory mixed-integer cuts. INFORMS J. Comput. 21, 641–649 (2009)

    MATH  MathSciNet  Google Scholar 

  89. W.J. Cook, J. Fonlupt, A. Schrijver, An integer analogue of Carathéodory’s theorem. J. Combin. Theory B 40, 63–70 (1986)

    MATH  MathSciNet  Google Scholar 

  90. W.J. Cook, R. Kannan, A. Schrijver, Chvátal closures for mixed integer programming problems. Math. Program. 47, 155–174 (1990)

    MATH  MathSciNet  Google Scholar 

  91. W.J. Cook, T. Rutherford, H.E. Scarf, D. Shallcross, An implementation of the generalized basis reduction algorithm for integer programming. ORSA J. Comput. 5, 206–212 (1993)

    MATH  MathSciNet  Google Scholar 

  92. G. Cornuéjols, Combinatorial Optimization: Packing and Covering. SIAM Monograph, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 74 (2001)

    Google Scholar 

  93. G. Cornuéjols, M.L. Fisher, G.L. Nemhauser, Location of bank accounts to optimize float: an analytic study of exact and approximate algorithms. Manag. Sci. 23, 789–810 (1977)

    MATH  Google Scholar 

  94. G. Cornuéjols, Y. Li, On the rank of mixed 0,1 polyhedra. Math. Program. A 91, 391–397 (2002)

    MATH  Google Scholar 

  95. G. Cornuéjols, Y. Li, A connection between cutting plane theory and the geometry of numbers. Math. Program. A 93, 123–127 (2002)

    MATH  Google Scholar 

  96. G. Cornuéjols, R. Tütüncü, Optimization Methods in Finance (Cambridge University Press, Cambridge, 2007)

    MATH  Google Scholar 

  97. A.M. Costa, A survey on Benders decomposition applied to fixed-charge network design problems. Comput. Oper. Res. 32, 1429–1450 (2005)

    MathSciNet  Google Scholar 

  98. H. Crowder, M.W. Padberg, Solving large-scale symmetric travelling salesman problems to optimality. Manag. Sci. 26, 495–509 (1980)

    MATH  MathSciNet  Google Scholar 

  99. H. Crowder, E. Johnson, M.W. Padberg, Solving large scale zero-one linear programming problems. Oper. Res. 31, 803–834 (1983)

    MATH  Google Scholar 

  100. R.J. Dakin, A tree-search algorithm for mixed integer programming problems. Comput. J. 8, 250–255 (1965)

    MATH  MathSciNet  Google Scholar 

  101. E. Danna, E. Rothberg, C. Le Pape, Exploring relaxation induced neighborhoods to improve MIP solutions. Math. Program. A 102, 71–90 (2005)

    MATH  Google Scholar 

  102. G.B. Dantzig, Maximization of a linear function of variables subject to linear inequalities, in Activity Analysis of Production and Allocation, ed. by T.C. Koopmans (Wiley, New York, 1951), pp. 339–347

    Google Scholar 

  103. G. Dantzig. R. Fulkerson, S. Johnson, Solution of a large-scale traveling-salesman problem. Oper. Res. 2, 393–410 (1954)

    MathSciNet  Google Scholar 

  104. G.B. Dantzig, P. Wolfe, Decomposition principle for linear programs. Oper. Res. 8, 101–111 (1960)

    MATH  Google Scholar 

  105. L. Danzer, B. Grünbaum, V. Klee, Helly’s theorem and its relatives, in Convexity, ed. by V. Klee (American Mathematical Society, Providence, 1963), pp. 101–180

    Google Scholar 

  106. S. Dash, S.S. Dey, O. Günlük, Two dimensional lattice-free cuts and asymmetric disjunctions for mixed-integer polyhedra. Math. Program. 135, 221–254 (2012)

    MATH  MathSciNet  Google Scholar 

  107. S. Dash, O. Günlük, A. Lodi, in On the MIR Closure of Polyhedra, IPCO 2007, ed. by M. Fischetti, D.P. Williamson. LNCS, Springer vol. 4513 (2007), pp. 337–351

    Google Scholar 

  108. R. Dechter, Constraint Processing (Morgan Kaufmann, San Francisco, 2003)

    Google Scholar 

  109. J.A. De Loera, J. Lee, P.N. Malkin, S. Margulies, Computing infeasibility certificates for combinatorial problems through Hilbert’s Nullstellensatz. J. Symb. Comput. 46, 1260–1283 (2011)

    MATH  Google Scholar 

  110. J.A. De Loera, R. Hemmecke, M. Köppe, Algebraic and Geometric Ideas in the Theory of Discrete Optimization. MOS-SIAM Series on Optimization, vol. 14 (2012)

    Google Scholar 

  111. R. de Wolf, Nondeterministic quantum query and communication complexities. SIAM J. Comput. 32, 681–699 (2003)

    MATH  MathSciNet  Google Scholar 

  112. A. Del Pia, R. Weismantel, Relaxations of mixed integer sets from lattice-free polyhedra. 4OR 10, 221–244 (2012)

    Google Scholar 

  113. A. Del Pia, R. Weismantel, On convergence in mixed integer programming. Math. Program. 135, 397–412 (2012)

    MATH  MathSciNet  Google Scholar 

  114. J. Desrosiers, F. Soumis, M. Desrochers, Routing with time windows by column generation. Networks 14, 545–565 (1984)

    MATH  Google Scholar 

  115. S.S. Dey, Q. Louveaux, Split rank of triangle and quadrilateral inequalities. Math. Oper. Res. 36, 432–461 (2011)

    MATH  MathSciNet  Google Scholar 

  116. S. S. Dey, D.A. Morán, On maximal S-free convex sets. SIAM J. Discrete Math. 25(1), 379–393 (2011)

    MATH  MathSciNet  Google Scholar 

  117. S.S. Dey, J.-P.P. Richard, Y. Li, L.A. Miller, On the extreme inequalities of infinite group problems. Math. Program. A 121, 145–170 (2010)

    MATH  MathSciNet  Google Scholar 

  118. S.S. Dey, L.A. Wolsey, Lifting Integer Variables in Minimal Inequalities Corresponding to Lattice-Free Triangles, IPCO 2008, Bertinoro, Italy. Lecture Notes in Computer Science, Springer, vol. 5035 (2008), pp. 463–475

    MathSciNet  Google Scholar 

  119. S.S. Dey, L.A. Wolsey, Constrained infinite group relaxations of MIPs. SIAM J. Optim. 20, 2890–2912 (2010)

    MATH  MathSciNet  Google Scholar 

  120. E.A. Dinic, Algorithm for solution of a problem of maximum flow in networks with power estimation. Soviet Math. Dokl. 11, 1277–1280 (1970)

    Google Scholar 

  121. J.-P. Doignon, Convexity in cristallographical lattices. J. Geom. 3, 71–85 (1973)

    MATH  MathSciNet  Google Scholar 

  122. M. Dyer, A. Frieze, R. Kannan, A random polynomial-time algorithm for approximating the volume of convex bodies. J. ACM 38, 1–17 (1991)

    MATH  MathSciNet  Google Scholar 

  123. J. Edmonds, Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)

    MATH  MathSciNet  Google Scholar 

  124. J. Edmonds, Maximum matching and a polyhedron with 0,1-vertices. J. Res. Natl. Bur. Stand. B 69, 125–130 (1965)

    MATH  MathSciNet  Google Scholar 

  125. J. Edmonds, Systems of distinct representatives and linear algebra. J. Res. Natl. Bur. Stand. B 71, 241–245 (1967)

    MATH  MathSciNet  Google Scholar 

  126. J. Edmonds, Submodular functions, matroids, and certain polyhedra, in Combinatorial Structures and Their Applications, ed. by R. Guy, H. Hanani, N. Sauer, J. Schönheim. (Gordon and Breach, New York, 1970), pp. 69–87

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  130. F. Eisenbrand, On the membership problem for the elementary closure of a polyhedron. Combinatorica 19, 297–300 (1999)

    MATH  MathSciNet  Google Scholar 

  131. F. Eisenbrand, G. Shmonin, Carathéodory bounds on integer cones. Oper. Res. Lett. 34, 564–568 (2006)

    MATH  MathSciNet  Google Scholar 

  132. F. Eisenbrand, A.S. Schulz, Bounds on the Chvátal rank of polytopes in the 0/1 cube. Combinatorica 23, 245–261 (2003)

    MATH  MathSciNet  Google Scholar 

  133. D. Erlenkotter, A dual-based procedure for uncapacitated facility location. Oper. Res. 26, 992–1009 (1978)

    MATH  MathSciNet  Google Scholar 

  134. T. Fahle, S. Shamberger, M. Sellmann, Symmetry Breaking, CP 2001. LNCS, vol. 2239 (2001), pp. 93–107

    Google Scholar 

  135. Gy. Farkas, On the applications of the mechanical principle of Fourier, Mathematikai és Természettudományi Értesotö 12, 457–472 (1894)

    Google Scholar 

  136. S. Fiorini, S. Massar, S. Pokutta, H.R. Tiwary, R. de Wolf, Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds, in STOC 2012 (2012)

    Google Scholar 

  137. S. Fiorini, V. Kaibel, K. Pashkovich, D.O. Theis Combinatorial bounds on the nonnegative rank and extended formulations. Discrete Math. 313, 67–83 (2013)

    MATH  MathSciNet  Google Scholar 

  138. M.L. Fischer, The Lagrangian relaxation method for solving integer programming problems. Manag. Sci. 27, 1–18 (1981)

    Google Scholar 

  139. M. Fischetti, F. Glover, A. Lodi, The feasibility pump. Math. Program. 104, 91–104 (2005)

    MATH  MathSciNet  Google Scholar 

  140. M. Fischetti, A. Lodi, Local branching. Math. Program. B 98, 23–47 (2003)

    MATH  MathSciNet  Google Scholar 

  141. M. Fischetti, A. Lodi, Optimizing over the first Chvátal closure. Math. Program. 110, 3–20 (2007)

    MATH  MathSciNet  Google Scholar 

  142. M. Fischetti, A. Lodi, A. Tramontani, On the separation of disjunctive cuts. Math. Program. A 128, 205–230 (2011)

    MATH  MathSciNet  Google Scholar 

  143. M. Fischetti, D. Salvagnin, C. Zanette, A note on the selection of Benders’ cuts. Math. Program. B 124, 175–182 (2010)

    MATH  MathSciNet  Google Scholar 

  144. R. Fortet, Applications de l’algèbre de Boole en recherche opérationnelle. Revue Française de Recherche Opérationnelle 4, 17–26 (1960)

    Google Scholar 

  145. J.B.J. Fourier, Solution d’une question particulière du calcul des inégalités. Nouveau Bulletin des Sciences par la Société Philomatique de Paris (1826), pp. 317–319

    Google Scholar 

  146. L.R. Ford Jr., D.R. Fulkerson, Maximal flow through a network. Can. J. Math. 8, 399–404 (1956)

    MATH  MathSciNet  Google Scholar 

  147. L.R. Ford Jr., D.R. Fulkerson, Flows in Networks (Princeton University Press, Princeton, 1962)

    MATH  Google Scholar 

  148. A. Frank, Connections in combinatorial optimization, in Oxford Lecture Series in Mathematics and Its Applications, vol. 38 (Oxford University Press, Oxford, 2011)

    Google Scholar 

  149. A. Frank, E. Tardos, An application of simultaneous Diophantine approximation in combinatorial optimization. Combinatorica 7, 49–65 (1987)

    MATH  MathSciNet  Google Scholar 

  150. R. M. Freund, J.B. Orlin, On the complexity of four polyhedral set containment problems. Math. Program. 33, 139–145 (1985)

    MATH  MathSciNet  Google Scholar 

  151. A.M. Frieze, M. Jerrum, Improved approximation algorithms for MAX k-CUT and MAX BISECTION. Algorithmica 18, 67–81 (1997)

    MATH  MathSciNet  Google Scholar 

  152. K. Fukuda, Frequently Asked Questions in Polyhedral Computation. Research Report, Department of Mathematics, and Institute of Theoretical Computer Science ETH Zurich, available online (2013)

    Google Scholar 

  153. K. Fukuda, Lecture: Polyhedral Computation. Research Report, Department of Mathematics, and Institute of Theoretical Computer Science ETH Zurich, available online (2004)

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  156. D.R Fulkerson, Blocking polyhedra, B Harris (Ed.), Graph Theory and Its Applications, Academic Press, New York 93–112 (1970)

    Google Scholar 

  157. D.R. Fulkerson, G.L. Nemhauser, L.E. Trotter, Two computationally difficult set covering problems that arise in computing the 1-width of incidence matrices of Steiner triples. Math. Program. Study 2, 72–81 (1974)

    Google Scholar 

  158. M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (W.H. Freeman and Co., New York, 1979)

    MATH  Google Scholar 

  159. R.S. Garfinkel, G. Nemhauser, Integer Programming (Wiley, New York, 1972)

    MATH  Google Scholar 

  160. C.F. Gauss, Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium (F. Perthes & J.H. Besser, Hamburg, 1809)

    Google Scholar 

  161. A.M. Geoffrion, Generalized Benders decomposition. J. Optim. Theory Appl. 10, 237–260 (1972)

    MATH  MathSciNet  Google Scholar 

  162. A.M. Geoffrion, Lagrangean relaxation for integer programming. Math. Program. Study 2, 82–114 (1974)

    MathSciNet  Google Scholar 

  163. A.M. Geoffrion, G.W. Graves, Multicommodity distribution design by Benders’ decomposition. Manag. Sci. 20, 822–844 (1974)

    MATH  MathSciNet  Google Scholar 

  164. A.M.H. Gerards, A short proof of Tutte’s characterization of totally unimodular matrices. Linear Algebra Appl. 114/115, 207–212 (1989)

    Google Scholar 

  165. A. Ghouila-Houri, Caractérisation des matrices totalement unimodulaires. Comptes Rendus Hebdomadaires des Scéances de l’Académie des Sciences (Paris) 254, 1192–1194 (1962)

    MATH  MathSciNet  Google Scholar 

  166. F.R. Giles, W.R. Pulleyblank, Total dual integrality and integer polyhedra. Linear Algebra Appl. 25, 191–196 (1979)

    MATH  MathSciNet  Google Scholar 

  167. P.C. Gilmore, Families of sets with faithful graph representation. IBM Research Note N.C., vol. 184 (Thomas J. Watson Research Center, Yorktown Heights, 1962)

    Google Scholar 

  168. P.C. Gilmore, R.E. Gomory, A linear programming approach to the cutting-stock problem. Oper. Res. 9, 849–859 (1961)

    MATH  MathSciNet  Google Scholar 

  169. M.X. Goemans, Worst-case comparison of valid inequalities for the TSP. Math. Program. 69, 335–349 (1995)

    MATH  MathSciNet  Google Scholar 

  170. M.X. Goemans, Smallest compact formulation for the permutahedron. Math. Program. Ser. A DOI 10.1007/s101007-014-0757-1 (2014)

    Google Scholar 

  171. M.X. Goemans, T. Rothvoß, Polynomiality for bin packing with a constant number of item types. arXiv:1307.5108 [cs.DS] (2013)

    Google Scholar 

  172. M.X. Goemans, L. Tunçel, When does the positive semidefiniteness constraint help in lifting procedures. Math. Oper. Res. 26, 796–815 (2001)

    MATH  MathSciNet  Google Scholar 

  173. M.X. Goemans, D.P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)

    MATH  MathSciNet  Google Scholar 

  174. J.L. Goffin, Variable metric relaxation methods, part II: the ellipsoid method. Math. Program. 30, 147–162 (1984)

    MATH  MathSciNet  Google Scholar 

  175. R.E. Gomory, Outline of an algorithm for integer solutions to linear programs. Bull. Am. Math. Soc. 64, 275–278 (1958)

    MATH  MathSciNet  Google Scholar 

  176. R.E. Gomory, An algorithm for the mixed integer problem. Tech. Report RM-2597 (The Rand Corporation, 1960)

    Google Scholar 

  177. R.E. Gomory, An algorithm for integer solutions to linear programs, in Recent Advances in Mathematical Programming, ed. by R.L. Graves, P. Wolfe (McGraw-Hill, New York, 1963), pp. 269–302

    Google Scholar 

  178. R.E. Gomory, Some polyhedra related to combinatorial problems. Linear Algebra Appl. 2, 451–558 (1969)

    MATH  MathSciNet  Google Scholar 

  179. R.E. Gomory, E.L. Johnson, Some continuous functions related to corner polyhedra I. Math. Program. 3, 23–85 (1972)

    MATH  MathSciNet  Google Scholar 

  180. R.E. Gomory, E.L. Johnson, T-space and cutting planes. Math. Program. 96, 341–375 (2003)

    MATH  MathSciNet  Google Scholar 

  181. J. Gouveia, P. Parrilo, R. Thomas, Theta bodies for polynomial ideals. SIAM J. Optim. 20, 2097–2118 (2010)

    MATH  MathSciNet  Google Scholar 

  182. J. Gouveia, P. Parrilo, R. Thomas, Lifts of convex sets and cone factorizations. Math. Oper. Res. 38, 248–264 (2013)

    MATH  MathSciNet  Google Scholar 

  183. M. Grötschel, Polyedrische Charackterisierungen kombinatorischer Optimierungsprobleme (Anton Hain, Meisenheim/Glan, 1977)

    Google Scholar 

  184. M. Grötschel, On the symmetric travelling salesman problem: solution of a 120-city problem. Math. Program. Study 12, 61–77 (1980)

    MATH  Google Scholar 

  185. M. Grötschel, M. Jünger, G. Reinelt, A cutting plane algorithm for the linear ordering problem. Oper. Res. 32, 1195–1220 (1984)

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  187. M. Grötschel, L. Lovász, A. Schrijver, Geometric methods in combinatorial optimization, in Progress in Combinatorial Optimization, ed. by W.R. Pulleyblank (Academic, Toronto, 1984), pp. 167–183

    Google Scholar 

  188. M. Grötschel, L. Lovász, A. Schrijver, Geometric Algorithms and Combinatorial Optimization (Springer, New York, 1988)

    MATH  Google Scholar 

  189. M. Grötschel, M.W. Padberg, On the symmetric travelling salesman problem I: inequalities. Math. Program. 16, (1979) 265–280

    MATH  Google Scholar 

  190. B. Grünbaum, Convex Polytopes (Wiley-Interscience, London, 1967)

    MATH  Google Scholar 

  191. Z. Gu, G.L. Nemhauser, M.W.P. Savelsbergh, Lifted flow covers for mixed 0–1 integer programs. Math. Program. 85, 439–467 (1999)

    MATH  MathSciNet  Google Scholar 

  192. Z. Gu, G.L. Nemhauser, M.W.P. Savelsbergh, Sequence independent lifting in mixed integer programming. J. Combin. Optim. 1, 109–129 (2000)

    MathSciNet  Google Scholar 

  193. C. Guéret, C. Prins, M. Servaux, Applications of Optimization with Xpress (Dash Optimization Ltd., London, 2002)

    Google Scholar 

  194. M. Guignard, S. Kim, Lagrangean decomposition for integer programming: theory and applications. RAIRO 21, 307–323 (1987)

    MATH  MathSciNet  Google Scholar 

  195. O. Günlük, Y. Pochet, Mixing mixed-integer inequalities. Math. Program. 90, 429–458 (2001)

    MATH  MathSciNet  Google Scholar 

  196. W. Harvey, Computing two-dimensional integer hulls. SIAM J. Comput. 28, 2285–2299 (1999)

    MATH  MathSciNet  Google Scholar 

  197. M. Held, R.M. Karp, The traveling-salesman problem and minimum spanning trees. Oper. Res. 18, 1138–1162 (1970)

    MATH  MathSciNet  Google Scholar 

  198. M. Held, R.M. Karp, The traveling-salesman problem and minimum spanning trees: part II. Math. Program. 1, 6–25 (1971)

    MATH  MathSciNet  Google Scholar 

  199. I. Heller, C.B. Tompkins, An extension of a theorem of Dantzig’s, in Linear Inequalities and Related Systems, ed. by H.W. Kuhn, A.W. Tucker (Princeton University Press, Princeton, 1956), pp. 247–254

    Google Scholar 

  200. Ch. Hermite, Extraits de lettres de M. Ch. Hermite à M. Jacobi sur différents objets de la théorie des nombres. Journal für dei reine und angewandte Mathematik 40, 261–277 (1850)

    MATH  Google Scholar 

  201. J.-B. Hiriart-Urruty, C. Lemaréchal. Fundamentals of Convex Analysis (Springer, New York, 2001)

    MATH  Google Scholar 

  202. D.S. Hirschberg, C.K. Wong, A polynomial algorithm for the knapsack problem in two variables. J. ACM 23, 147–154 (1976)

    MATH  MathSciNet  Google Scholar 

  203. A.J. Hoffman, A generalization of max-flow min-cut. Math. Program. 6, 352–259 (1974)

    MATH  Google Scholar 

  204. A.J. Hoffman, J.B. Kruskal, Integral boundary points of polyhedra, in Linear Inequalities and Related Systems, ed. by H.W. Kuhn, A.W. Tucker (Princeton University Press, Princeton, 1956), pp. 223–246

    Google Scholar 

  205. J.N. Hooker, Needed: an empirical science of algorithms. Oper. Res. 42, 201–212 (1994)

    MATH  Google Scholar 

  206. J. Hooker, Integrated Methods for Optimization. International Series in Operations Research and Management Science (Springer, New York, 2010)

    Google Scholar 

  207. R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 2013)

    MATH  Google Scholar 

  208. C.A.J. Hurkens, Blowing up convex sets in the plane. Linear Algebra Appl. 134, 121–128 (1990)

    MATH  MathSciNet  Google Scholar 

  209. S. Iwata, L. Fleischer, S. Fujishige, A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions. J. ACM 48, 761–777 (2001)

    MATH  MathSciNet  Google Scholar 

  210. R.G. Jeroslow, There cannot be any algorithm for integer programming with quadratic constraints. Oper. Res. 21, 221–224 (1973)

    MATH  MathSciNet  Google Scholar 

  211. R.G. Jeroslow, Representability in mixed integer programming, I: characterization results. Discrete Appl. Math. 17, 223–243 (1987)

    MATH  MathSciNet  Google Scholar 

  212. R.G Jeroslow, On defining sets of vertices of the hypercube by linear inequalities. Discrete Math. 11, 119–124 (1975)

    MATH  MathSciNet  Google Scholar 

  213. R.G Jeroslow, J.K. Lowe, Modelling with integer variables. Math. Program. Stud. 22, 167–184 (1984)

    MATH  MathSciNet  Google Scholar 

  214. F. John, Extremum problems with inequalities as subsidiary conditions, in Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948 (Interscience Publishers, New York, 1948), pp. 187–204

    Google Scholar 

  215. E.L. Johnson, On the group problem for mixed integer programming. Math. Program. Study 2, 137–179 (1974)

    Google Scholar 

  216. E.L. Johnson, Characterization of facets for multiple right-hand choice linear programs. Math. Program. Study 14, 112–142 (1981)

    MATH  Google Scholar 

  217. M. Jünger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, L. Wolsey (eds.), 50 Years of Integer Programming 1958–2008 (Springer, Berlin, 2010)

    MATH  Google Scholar 

  218. M. Jünger, D. Naddef (eds.), Computational Combinatorial Optimization. Optimal or provably near-optimal solutions. Lecture Notes in Computer Science, vol. 2241 (Springer, Berlin, 2001)

    Google Scholar 

  219. V. Kaibel, Extended formulations in combinatorial optimization. Optima 85, 2–7 (2011)

    Google Scholar 

  220. V. Kaibel, K. Pashkovich, Constructing extended formulations from reflection relations, in Proceedings of IPCO XV O. Günlük, ed. by G. Woeginger. Lecture Notes in Computer Science, vol. 6655 (Springer, Berlin, 2011), pp. 287–300

    Google Scholar 

  221. V. Kaibel, K. Pashkovich, D.O. Theis, Symmetry matters for sizes of extended formulations. SIAM J. Discrete Math. 26(3), 1361–1382 (2012)

    MATH  MathSciNet  Google Scholar 

  222. V. Kaibel, M.E. Pfetsch, Packing and partitioning orbitopes. Math. Program. 114, 1–36 (2008)

    MATH  MathSciNet  Google Scholar 

  223. V. Kaibel, S. Weltge, A short proof that the extension complexity of the correlation polytope grows exponentially. arXiv:1307.3543 (2013)

    Google Scholar 

  224. V. Kaibel, S. Weltge, Lower bounds on the sizes of integer programs without additional variables. arXiv:1311.3255 (2013)

    Google Scholar 

  225. R. Kannan, A polynomial algorithm for the two-variable integer programming problem. J. ACM 27, 118–122 (1980)

    MATH  Google Scholar 

  226. R. Kannan, Improved algorithms for integer programming and related problems, in Proceedings of the 15th Annual ACM Symposium on Theory of Computing (STOC-83) (1983), pp. 193–206

    Google Scholar 

  227. R. Kannan, Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12, 415–440 (1987)

    MATH  MathSciNet  Google Scholar 

  228. R. Kannan, A. Bachem, Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM J. Comput. 8, 499–507 (1979)

    MATH  MathSciNet  Google Scholar 

  229. N. Karmarkar, A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–395 (1984)

    MATH  MathSciNet  Google Scholar 

  230. D.R. Karger, Global min-cuts in RNC, and other ramifications of a simple min-cut algorithm, in Proceedings of SODA (1993), pp. 21–30

    Google Scholar 

  231. D.R. Karger, R. Motwani, M. Sudan, Approximate graph coloring by semidefinite programming. J. ACM 45, 246–265 (1998)

    MATH  MathSciNet  Google Scholar 

  232. R.M. Karp, Reducubility among combinatorial problems, in Complexity of Computer Computations (Plenum Press, New York, 1972), pp. 85–103

    Google Scholar 

  233. R.M. Karp, C.H. Papadimitriou, On linear characterizations of combinatorial optimization problems. SIAM J. Comput. 11, 620–632 (1982)

    MATH  MathSciNet  Google Scholar 

  234. H. Kellerer, U. Pferschy, D. Pisinger, Knapsack Problems (Springer, Berlin, 2004)

    MATH  Google Scholar 

  235. L.G. Khachiyan, A polynomial algorithm in linear programming. Soviet Math. Dokl. 20, 191–194 (1979)

    MATH  Google Scholar 

  236. L. Khachiyan, E. Boros, K. Borys, K. Elbassioni, V. Gurvich, Generating all vertices of a polyhedron is hard. Discrete Comput. Geom. 39, 174–190 (2008)

    MATH  MathSciNet  Google Scholar 

  237. A. Khinchine, A quantitative formulation of Kronecker’s theory of approximation (in russian). Izvestiya Akademii Nauk SSR Seriya Matematika 12, 113–122 (1948)

    Google Scholar 

  238. F. Kilinc-Karzan, G.L. Nemhauser, M.W.P. Savelsbergh, Information-based branching schemes for binary linear mixed integer problems. Math. Program. Comput. 1, 249–293 (2009)

    MathSciNet  Google Scholar 

  239. D. Klabjan, G.L. Nemhauser, C. Tovey, The complexity of cover inequality separation. Oper. Res. Lett. 23, 35–40 (1998)

    MATH  MathSciNet  Google Scholar 

  240. V. Klee, G.J. Minty, How good is the simplex algorithm? in Inequalities, III, ed. by O. Shisha (Academic, New York, 1972), pp. 159–175

    Google Scholar 

  241. M. Köppe, Q. Louveaux, R. Weismantel, Intermediate integer programming representations using value disjunctions. Discrete Optim. 5, 293–313 (2008)

    MATH  MathSciNet  Google Scholar 

  242. M. Köppe, R. Weismantel, A mixed-integer Farkas lemma and some consequences. Oper. Res. Lett. 32, 207–211 (2004)

    MATH  MathSciNet  Google Scholar 

  243. B. Korte, J. Vygen, Combinatorial Optimization: Theory and Algorithms (Springer, Berlin/Hidelberg, 2000)

    Google Scholar 

  244. J.B. Kruskal Jr., On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Am. Math. Soc. 7, 48–50 (1956)

    MATH  MathSciNet  Google Scholar 

  245. H.W. Kuhn, The Hungarian method for the assignment problem. Naval Res. Logistics Q. 2, 83–97 (1955)

    Google Scholar 

  246. A.H. Land, A.G. Doig, An automatic method of solving discrete programming problems. Econometrica 28, 497–520 (1960)

    MATH  MathSciNet  Google Scholar 

  247. J.B. Lasserre, An Explicit Exact SDP Relaxation for Nonlinear 0–1 Programs. Lecture Notes in Computer Science, vol. 2081 (2001), pp. 293–303

    MathSciNet  Google Scholar 

  248. J.B. Lasserre, Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)

    MATH  MathSciNet  Google Scholar 

  249. M. Laurent, A comparison of the Sherali-Adams, Lovász-Schrijver and Lasserre relaxations for 0–1 programming. SIAM J. Optim. 28, 345–375 (2003)

    Google Scholar 

  250. M. Laurent, F. Rendl, Semidefinite programming and integer programming, in Handbook on Discrete Optimization, ed. by K. Aardal, G.L. Nemhauser, R. Weimantel (Elsevier, Amsterdam, 2005), pp. 393–514

    Google Scholar 

  251. E. L. Lawler, Covering problems: duality relations and a method of solution. SIAM J. Appl. Math. 14, 1115–1132 (1966)

    MATH  MathSciNet  Google Scholar 

  252. E. L. Lawler, Combinatorial Optimization: Networks and Matroids (Holt, Rinehart and Winston, New York, 1976)

    MATH  Google Scholar 

  253. E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys (eds.), The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization (Wiley, New York, 1985)

    MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  255. A.K. Lenstra, H.W. Lenstra, L. Lovász, Factoring polynomials with rational coefficients. Math. Ann. 261, 515–534 (1982)

    MATH  MathSciNet  Google Scholar 

  256. H.W. Lenstra, Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983)

    MATH  MathSciNet  Google Scholar 

  257. J.T. Linderoth, M.W.P. Savelsbergh, A computational study of search strategies for mixed integer programming. INFORMS J. Comput. 11, 173–187 (1999)

    MATH  MathSciNet  Google Scholar 

  258. Q. Louveaux, L.A. Wolsey, Lifting, superadditivity, mixed integer rounding and single node flow sets revisited. 4OR 1, 173–207 (2003)

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  261. L. Lovász, Geometry of numbers and integer programming, in Mathematical Programming: Recent Developments and Applications, ed. by M. Iri, K. Tanabe (Kluwer, Dordrecht, 1989), pp. 177–201

    Google Scholar 

  262. L. Lovász, M.D. Plummer, Matching Theory (Akadémiai Kiadó, Budapest, 1986) [Also: North Holland Mathematics Studies, vol. 121 (North Holland, Amsterdam)]

    MATH  Google Scholar 

  263. L. Lovász, H.E. Scarf, The generalized basis reduction algorithm. Math. Oper. Res. 17, 751–764 (1992)

    MATH  MathSciNet  Google Scholar 

  264. L. Lovász, A. Schrijver, Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1, 166–190 (1991)

    MATH  MathSciNet  Google Scholar 

  265. T.L. Magnanti, R.T. Wong, Accelerated Benders decomposition: algorithmic enhancement and model selection criteria. Oper. Res. 29, 464–484 (1981)

    MATH  MathSciNet  Google Scholar 

  266. H. Marchand, L.A. Wolsey, Aggregation and mmixed integer rounding to solve MIPs. Oper. Res. 49, 363–371 (2001)

    MATH  MathSciNet  Google Scholar 

  267. F. Margot, Pruning by isomorphism in branch-and-cut. Math. Program. 94, 71–90 (2002)

    MATH  MathSciNet  Google Scholar 

  268. S. Martello, P. Toth, Knapsack Problems: Algorithms and Computer Implementations (Wiley, Chichester, 1990)

    MATH  Google Scholar 

  269. R.K. Martin, Generating alternative mixed integer programming models using variable definition. Oper. Res. 35, 820–831 (1987)

    MATH  MathSciNet  Google Scholar 

  270. R.K. Martin, Using separation algorithms to generate mixed integer model reformulations. Oper. Res. Lett. 10(3), 119–128 (1991)

    MATH  MathSciNet  Google Scholar 

  271. R.K. Martin, R.L. Rardin, B.A. Campbell, Polyhedral characterization of discrete dynamic programming. Oper. Res. 38, 127–138 (1990)

    MATH  MathSciNet  Google Scholar 

  272. J.F. Maurras, Bon algorithmes, vieilles idées, Note E.d.F. HR 32.0320 (1978)

    Google Scholar 

  273. J.F. Maurras, K. Truemper, M. Agkül, Polynomial algorithms for a class of linear programs. Math. Program. 21, 121–136 (1981)

    MATH  Google Scholar 

  274. C.C. McGeogh, Experimental analysis of algorithms. Notices Am. Math. Assoc. 48, 204–311 (2001)

    Google Scholar 

  275. B.D. McKay, Practical graph isomorphism. Congressus Numerantium 30, 45–87 (1981)

    MathSciNet  Google Scholar 

  276. R.R. Meyer, On the existence of optimal solutions to integer and mixed integer programming problems. Math. Program. 7, 223–235 (1974)

    MATH  Google Scholar 

  277. D. Micciancio, The shortest vector in a lattice is hard to approximate to within some constant, in Proceedings of the 39th Annual Symposium on Foundations of Computer Science (FOCS-98) (1998), pp. 92–98

    Google Scholar 

  278. C.E. Miller, A.W. Tucker, R.A. Zemlin, Integer programming formulation of traveling salesman problems. J. ACM 7, 326–329 (1960)

    MATH  MathSciNet  Google Scholar 

  279. H. Minkowski, Geometrie der Zahlen (Erste Lieferung) (Teubner, Leipzig, 1896)

    Google Scholar 

  280. T.S. Motzkin, H. Raiffa, G.L. Thompson, R.M. Thrall, The double description method, in Contributions to Theory of Games, vol. 2, ed. by H.W. Kuhn, A.W. Tucker (Princeton University Press, Princeton, 1953)

    Google Scholar 

  281. J. Munkres, Algorithms for the assignment and transportation problems. J. SIAM 5, 32–38 (1957)

    MATH  MathSciNet  Google Scholar 

  282. H. Nagamochi, T. Ibaraki, Computing edge-connectivity in multiple and capacitated graphs. SIAM J. Discrete Math. 5, 54–66 (1992)

    MATH  MathSciNet  Google Scholar 

  283. G.L. Nemhauser, L.E. Trotter Jr., Properties of vertex packing and independence system polyhedra. Math. Program. 6, 48–61 (1974)

    MATH  MathSciNet  Google Scholar 

  284. G.L. Nemhauser, L.E. Trotter Jr., Vertex packings: structural properties and algorithms. Math. Program. 8, 232–248 (1975)

    MATH  MathSciNet  Google Scholar 

  285. G.L. Nemhauser, L.A. Wolsey, Integer and Combinatorial Optimization (Wiley, New York, 1988)

    MATH  Google Scholar 

  286. G.L. Nemhauser, L.A. Wolsey, A recursive procedure to generate all cuts for 0–1 mixed integer programs. Math. Program. 46, 379–390 (1990)

    MATH  MathSciNet  Google Scholar 

  287. Y.E. Nesterov, Smooth minimization of non-smooth functions. Math. Program. A 103, 127–152 (2005)

    MATH  MathSciNet  Google Scholar 

  288. Y.E. Nesterov, Semidefinite relaxation and nonconvex quadratic optimization. Optim. Methods Softw. 12, 1–20 (1997)

    MathSciNet  Google Scholar 

  289. Y.E. Nesterov, A.S. Nemirovski, Self-concordant functions and polynomial time methods in convex programming. Technical report, Central Economical and Mathematical Institute, U.S.S.R (Academy of Science, Moscow, 1990)

    Google Scholar 

  290. Y.E. Nesterov, A.S. Nemirovski, Conic formulation of a convex programming problem and duality. Optim. Methods Softw. 1, 95–115 (1992)

    Google Scholar 

  291. Y.E. Nesterov, A.S. Nemirovski, Interior Point Polynomial Algorithms in Convex Programming (SIAM, Philadelphia, 1994)

    MATH  Google Scholar 

  292. J. Ostrowski, J.T. Linderoth, F. Rossi, S. Smriglio, Solving large Steiner triple covering problems. Oper. Res. Lett. 39, 127–131 (2011)

    MATH  MathSciNet  Google Scholar 

  293. J. Ostrowski, J. Linderoth, F. Rossi, S. Smriglio, Orbital branching. Math. Program. 126, 147–178 (2011)

    MATH  MathSciNet  Google Scholar 

  294. J.H. Owen, S. Mehrotra, A disjunctive cutting plane procedure for general mixed-integer linear programs. Math. Program. A 89, 437–448 (2001)

    MATH  MathSciNet  Google Scholar 

  295. J.H. Owen, S. Mehrotra, On the value of binary expansions for general mixed-integer linear programs. Oper. Res. 50, 810–819 (2002)

    MATH  MathSciNet  Google Scholar 

  296. J. Oxley, Matroid Theory (Oxford University Press, New York, 2011)

    MATH  Google Scholar 

  297. M.W. Padberg, On the facial structure of set packing polyhedra. Math. Program. 5, 199–215 (1973)

    MATH  MathSciNet  Google Scholar 

  298. M.W. Padberg, A note on zero-one programming. Oper. Res. 23, 833–837 (1975)

    MATH  MathSciNet  Google Scholar 

  299. M.W. Padberg, M.R. Rao, The Russian method for linear programming III: bounded integer programming. Research Report 81-39, Graduate School of Business Administration, New York University (1981)

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  301. M.W. Padberg, G. Rinaldi, Optimization of a 532-city symmetric traveling salesman problem by branch and cut. Oper. Res. Lett. 6, 1–7 (1987)

    MATH  MathSciNet  Google Scholar 

  302. M.W. Padberg, G. Rinaldi, A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems. SIAM Rev. 33, 60–100 (1991)

    MATH  MathSciNet  Google Scholar 

  303. M. Padberg, T.J. Van Roy, L.A. Wolsey, Valid linear inequalities for fixed charge problems. Oper. Res. 33, 842–861 (1985)

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  305. C.H. Papadimitriou, On the complexity of integer programming. J. ACM 28, 765–768 (1981)

    MATH  MathSciNet  Google Scholar 

  306. J. Patel, J.W. Chinneck, Active-constraint variable ordering for faster feasibility of mixed integer linear programs. Math. Program. 110, 445–474 (2007)

    MATH  MathSciNet  Google Scholar 

  307. J. Petersen, Die Theorie der regulären graphs. Acta Matematica 15, 193–220 (1891)

    MATH  Google Scholar 

  308. Y. Pochet, L.A. Wolsey, Polyhedra for lot-sizing with Wagner–Whitin costs. Math. Program. 67, 297–324 (1994)

    MATH  MathSciNet  Google Scholar 

  309. Y. Pochet, L.A. Wolsey, Production Planning by Mixed-Integer Programming. Springer Series in Operations Research and Financial Engineering (Springer, New York, 2006)

    Google Scholar 

  310. B.T. Poljak, A general method for solving extremum problems. Soviet Math. Dokl. 8, 593–597 (1967)

    Google Scholar 

  311. C.H. Papadimitriou, M. Yannakakis, On recognizing integer polyhedra. Combinatorica 10, 107–109 (1990)

    MATH  MathSciNet  Google Scholar 

  312. M. Queyranne, A.S. Schulz, Polyhedral approaches to machine scheduling. Preprint (1994)

    Google Scholar 

  313. A. Razborov, On the distributional complexity of disjointness. Theor. Comput. Sci. 106(2), 385–390 (1992)

    MATH  MathSciNet  Google Scholar 

  314. J. Renegar, A polynomial-time algorithm based on Newton’s method for linear programming. Math. Program. 40, 59–93 (1988)

    MATH  MathSciNet  Google Scholar 

  315. J.-P.P. Richard, S.S. Dey (2010). The group-theoretic approach in mixed integer programming, in 50 Years of Integer Programming 1958–2008, ed. by M. Jünger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, L. Wolsey (Springer, New York, 2010), pp. 727–801

    Google Scholar 

  316. R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1969)

    Google Scholar 

  317. T. Rothvoß, Some 0/1 polytopes need exponential size extended formulations. Math. Program. A 142, 255–268 (2012)

    Google Scholar 

  318. T. Rothvoß, The matching polytope has exponential extension complexity, in Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC 2014), (2014), pp. 263–272

    Google Scholar 

  319. T. Rothvoß, L. Sanitá, 0 − 1 polytopes with quadratic Chvátal rank, in Proceedings of the 16th IPCO Conference. Lecture Notes in Computer Science, vol. 7801 (Springer, New York, 2013)

    Google Scholar 

  320. J.-S. Roy, Reformulation of bounded integer variables into binary variables to generate cuts. Algorithmic Oper. Res. 2, 810–819 (2007)

    Google Scholar 

  321. M.P.W. Savelsbergh, Preprocessing and probing techniques for mixed integer programming problems. ORSA J. Comput. 6, 445–454 (1994)

    MATH  MathSciNet  Google Scholar 

  322. H.E. Scarf, An observation on the structure of production sets with indivisibilities. Proc. Natl. Acad. Sci. USA 74, 3637–3641 (1977)

    MATH  MathSciNet  Google Scholar 

  323. A. Schrijver, On cutting planes. Ann. Discrete Math. 9, 291–296 (1980)

    MATH  MathSciNet  Google Scholar 

  324. A. Schrijver, On total dual integrality. Linear Algebra Appl. 38, 27–32 (1981)

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  326. A. Schrijver, A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Combin. Theory Ser. B 80, 346–355 (2000)

    MATH  MathSciNet  Google Scholar 

  327. A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency (Springer, Berlin, 2003)

    Google Scholar 

  328. Á. Seress, Permutation Group Algorithms, Cambridge Tracts in Mathematics, vol. 152 (Cambridge University Press, Cambridge, 2003)

    Google Scholar 

  329. P.D. Seymour, Decomposition of regular matroids. J. Combin. Theory B 28, 305–359 (1980)

    MATH  MathSciNet  Google Scholar 

  330. H. Sherali, W. Adams, A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discrete Math. 3, 311–430 (1990)

    MathSciNet  Google Scholar 

  331. H. Sherali, W. Adams, A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems, Chap. 4 (Kluwer Academic Publishers, Norwell, 1999)

    Google Scholar 

  332. N. Z. Shor, Cut-off method with space extension in convex programming problems. Cybernetics 13, 94–96 (1977)

    Google Scholar 

  333. M. Stoer, F. Wagner, A simple min-cut algorithm. J. ACM 44, 585–591 (1997)

    MATH  MathSciNet  Google Scholar 

  334. E. Tardos, A strongly polynomial algorithm to solve combinatorial linear programs. Oper. Res. 34, 250–256 (1986)

    MATH  MathSciNet  Google Scholar 

  335. R.E. Tarjan, Depth-first search and linear graph algorithms. SIAM J. Comput. 1, 146–160 (1972)

    MATH  MathSciNet  Google Scholar 

  336. S. Tayur, R.R. Thomas, N.R. Natraj, An algebraic geometry algorithm for scheduling in presence of setups and correlated demands. Math. Program. 69, 369–401 (1995)

    MATH  MathSciNet  Google Scholar 

  337. P. Toth, D. Vigo, The Vehicle Routing Problem. Monographs on Discrete Mathematics and Applications (SIAM, Philadelphia, 2001)

    Google Scholar 

  338. K. Truemper, Matroid Decomposition (Academic, Boston, 1992)

    MATH  Google Scholar 

  339. W.T. Tutte, A homotopy theorem for matroids I, II. Trans. Am. Math. Soc. 88, 905–917 (1958)

    MathSciNet  Google Scholar 

  340. T.J. Van Roy, L.A. Wolsey, Solving mixed integer programming problems using automatic reformulation. Oper. Res. 35, 45–57 (1987)

    MATH  MathSciNet  Google Scholar 

  341. M. Van Vyve, The continuous mixing polyhedron. Math. Oper. Res. 30, 441–452 (2005)

    MATH  MathSciNet  Google Scholar 

  342. F. Vanderbeck, L.A. Wolsey, Reformulation and decomposition of integer programs, in 50 Years of Integer Programming 1958–2008, ed. by M. Jünger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, L. Wolsey (Springer, New York, 2010), pp. 431–502

    Google Scholar 

  343. R.J. Vanderbei, Linear Programming: Foundations and Extentions, 3rd edn. (Springer, New York, 2008)

    Google Scholar 

  344. S. Vavasis, On the complexity of nonnegative matrix factorization. SIAM J. Optim. 20, 1364–1377 (2009)

    MATH  MathSciNet  Google Scholar 

  345. V.V. Vazirani, Approximation Algorithms (Springer, Berlin, 2003)

    Google Scholar 

  346. J.P. Vielma, A constructive characterization of the split closure of a mixed integer linear program. Oper. Res. Lett. 35, 29–35 (2007)

    MATH  MathSciNet  Google Scholar 

  347. J.P. Vielma, Mixed integer linear programming formulation techniques to appear in SIAM Review (2014)

    Google Scholar 

  348. H. Weyl, The elementary theory of convex polyhedra, in Contributions to the Theory of Games I, ed. by H.W. Kuhn, A.W. Tucker (Princeton University Press, Princeton, 1950), pp. 3–18

    Google Scholar 

  349. D.P. Williamson, D.B. Shmoys, The Design of Approxiamtion Algorithms (Cambridge University Press, Cambridge, 2011)

    Google Scholar 

  350. L.A. Wolsey, Further facet generating procedures for vertex packing polytopes. Math. Program. 11, 158–163 (1976)

    MATH  MathSciNet  Google Scholar 

  351. L.A. Wolsey, Valid inequalities and superadditivity for 0–1 integer programs. Math. Oper. Res. 2, 66–77 (1977)

    MATH  MathSciNet  Google Scholar 

  352. L.A. Wolsey, Heuristic analysis, linear programming, and branch and bound. Math. Program. Stud. 13, 121–134 (1980)

    MATH  MathSciNet  Google Scholar 

  353. L.A. Wolsey, Integer Programming (Wiley, New York, 1999)

    Google Scholar 

  354. R.T. Wong, Dual ascent approach for Steiner tree problems on directed graphs. Math. Program. 28, 271–287 (1984)

    MATH  Google Scholar 

  355. M. Yannakakis, Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43, 441–466 (1991)

    MATH  MathSciNet  Google Scholar 

  356. D. B. Yudin, A. S. Nemirovski, Evaluation of the information complexity of mathematical programming problems. Ekonomika i Matematicheskie Metody 12, 128–142 (1976) (in Russian). English Translation: Matekon 13, 3–45 (1976)

    Google Scholar 

  357. G.M. Ziegler, Lectures on Polytopes (Springer, New York, 1995)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Padova, Padova, Italy

    Michele Conforti

  2. Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA

    Gérard Cornuéjols

  3. Department of Management, London School of Economics and Political Science, London, UK

    Giacomo Zambelli

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  1. Michele Conforti
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  3. Giacomo Zambelli
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Conforti, M., Cornuéjols, G., Zambelli, G. (2014). Enumeration. In: Integer Programming. Graduate Texts in Mathematics, vol 271. Springer, Cham. https://doi.org/10.1007/978-3-319-11008-0_9

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