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Valid Inequalities for Structured Integer Programs

  • Michele Conforti
  • Gérard Cornuéjols
  • Giacomo Zambelli
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 271)

Abstract

In Chaps.  5 and  6 we have introduced several classes of valid inequalities that can be used to strengthen integer programming formulations in a cutting plane scheme. All these valid inequalities are “general purpose,” in the sense that their derivation does not take into consideration the structure of the specific problem at hand. Many integer programs have an underlying combinatorial structure, which can be exploited to derive “strong” valid inequalities, where the term “strong” typically refers to the fact that the inequality is facet-defining for the convex hull of feasible solutions.

Bibliography

  1. [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)zbMATHMathSciNetGoogle Scholar
  2. [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. [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)zbMATHMathSciNetGoogle Scholar
  4. [4]
    K. Aardal, R. Weismantel, L.A. Wolsey, Non-standard approaches to integer programming. Discrete Appl. Math. 123, 5–74 (2002)zbMATHMathSciNetGoogle Scholar
  5. [5]
    T. Achterberg, Constraint Integer Programming. Ph.D. thesis, ZIB, Berlin, 2007Google Scholar
  6. [6]
    T. Achterberg, T. Berthold, Improving the feasibility pump. Discrete Optim. 4, 77–86 (2007)zbMATHMathSciNetGoogle Scholar
  7. [7]
    T. Achterberg, T. Koch, A. Martin, Branching rules revisited. Oper. Res. Lett. 33, 42–54 (2005)zbMATHMathSciNetGoogle Scholar
  8. [8]
    T. Achterberg, T. Koch, A. Martin, MIPLIB 2003. Oper. Res. Lett. 34, 361–372 (2006)zbMATHMathSciNetGoogle Scholar
  9. [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. 10–19Google Scholar
  10. [10]
    F. Alizadeh, Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim. 5, 13–51 (1995)zbMATHMathSciNetGoogle Scholar
  11. [11]
    K. Andersen, G. Cornuéjols, Y. Li, Split closure and intersection cuts. Math. Program. A 102, 457–493 (2005)zbMATHGoogle Scholar
  12. [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. 1–15MathSciNetGoogle Scholar
  13. [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. [14]
    S. Arora, B. Barak, Complexity Theory: A Modern Approach (Cambridge University Press, Cambridge, 2009)Google Scholar
  15. [15]
    A. Atamtürk, Strong formulations of robust mixed 0–1 programming. Math. Program. 108, 235–250 (2006)zbMATHMathSciNetGoogle Scholar
  16. [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)zbMATHGoogle Scholar
  17. [17]
    R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows (Prentice Hall, New Jersey, 1993)zbMATHGoogle Scholar
  18. [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)zbMATHMathSciNetGoogle Scholar
  19. [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. [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)zbMATHMathSciNetGoogle Scholar
  21. [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. 19–28Google Scholar
  22. [22]
    E. Balas, Intersection cuts—a new type of cutting planes for integer programming. Oper. Res. 19, 19–39 (1971)zbMATHMathSciNetGoogle Scholar
  23. [23]
    E. Balas, Integer programming and convex analysis: intersection cuts from outer polars. Math. Program. 2 330–382 (1972)zbMATHMathSciNetGoogle Scholar
  24. [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. [25]
    E. Balas, Facets of the knapsack polytope. Math. Program. 8, 146–164 (1975)zbMATHMathSciNetGoogle Scholar
  26. [26]
    E. Balas, Disjunctive programming and a hierarchy of relaxations for discrete optimization problems. SIAM J. Algebr. Discrete Methods 6, 466–486 (1985)zbMATHMathSciNetGoogle Scholar
  27. [27]
    E. Balas, A modified lift-and-project procedure. Math. Program. 79, 19–31 (1997)zbMATHMathSciNetGoogle Scholar
  28. [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)zbMATHMathSciNetGoogle Scholar
  29. [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)zbMATHGoogle Scholar
  30. [30]
    E. Balas, S. Ceria, G. Cornuéjols, R.N. Natraj, Gomory cuts revisited. Oper. Res. Lett. 19, 1–9 (1996)zbMATHMathSciNetGoogle Scholar
  31. [31]
    E. Balas, R. Jeroslow, Strengthening cuts for mixed integer programs. Eur. J. Oper. Res. 4, 224–234 (1980)zbMATHMathSciNetGoogle Scholar
  32. [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)zbMATHMathSciNetGoogle Scholar
  33. [33]
    E. Balas, W.R. Pulleyblank, The perfectly matchable subgraph polytope of an arbitrary graph. Combinatorica 9, 321–337 (1989)zbMATHMathSciNetGoogle Scholar
  34. [34]
    E. Balas, A. Saxena, Optimizing over the split closure. Math. Program. 113, 219–240 (2008)zbMATHMathSciNetGoogle Scholar
  35. [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)zbMATHMathSciNetGoogle Scholar
  36. [36]
    F. Barahona, R. Anbil, The volume algorithm: producing primal solutions with a subgradient method. Math. Program. 87, 385–399 (2000)zbMATHMathSciNetGoogle Scholar
  37. [37]
    I. Barany, T.J. Van Roy, L.A. Wolsey, Uncapacitated lot-sizing: the convex hull of solutions. Math. Program. 22, 32–43 (1984)zbMATHGoogle Scholar
  38. [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)zbMATHMathSciNetGoogle Scholar
  39. [39]
    A. Barvinok, A Course in Convexity. Graduate Studies in Mathematics, vol. 54 (American Mathematical Society, Providence, 2002)Google Scholar
  40. [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)zbMATHGoogle Scholar
  41. [41]
    A. Basu, M. Conforti, G. Cornuéjols, G. Zambelli, Maximal lattice-free convex sets in linear subspaces. Math. Oper. Res. 35, 704–720 (2010)zbMATHMathSciNetGoogle Scholar
  42. [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)zbMATHMathSciNetGoogle Scholar
  43. [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)zbMATHMathSciNetGoogle Scholar
  44. [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. [45]
    D.E. Bell, A theorem concerning the integer lattice. Stud. Appl. Math. 56, 187–188 (1977)zbMATHGoogle Scholar
  46. [46]
    R. Bellman, Dynamic Programming (Princeton University Press, Princeton, 1957)zbMATHGoogle Scholar
  47. [47]
    J.F. Benders, Partitioning procedures for solving mixed variables programming problems. Numerische Mathematik 4, 238–252 (1962)zbMATHMathSciNetGoogle Scholar
  48. [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)zbMATHGoogle Scholar
  49. [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. [50]
    C. Berge, Two theorems in graph theory. Proc. Natl. Acad. Sci. USA 43, 842–844 (1957)zbMATHMathSciNetGoogle Scholar
  51. [51]
    D. Bertsimas, R. Weismantel, Optimization over Integers (Dynamic Ideas, Belmont, 2005)Google Scholar
  52. [52]
    D. Bienstock, M. Zuckerberg, Subset algebra lift operators for 0–1 integer programming. SIAM J. Optim. 15, 63–95 (2004)zbMATHMathSciNetGoogle Scholar
  53. [53]
    L.J. Billera, A. Sarangarajan, All 0,1 polytopes are traveling salesman polytopes. Combinatorica 16, 175–188 (1996)zbMATHMathSciNetGoogle Scholar
  54. [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. [55]
    J. R. Birge, F. Louveaux, Introduction to Stochastic Programming (Springer, New York, 2011)zbMATHGoogle Scholar
  56. [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. [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. 309–326Google Scholar
  58. [58]
    P. Bonami, On optimizing over lift-and-project closures. Math. Program. Comput. 4, 151–179 (2012)zbMATHMathSciNetGoogle Scholar
  59. [59]
    P. Bonami, M. Conforti, G. Cornuéjols, M. Molinaro, G. Zambelli, Cutting planes from two-term disjunctions. Oper. Res. Lett. 41, 442–444 (2013)zbMATHMathSciNetGoogle Scholar
  60. [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)zbMATHMathSciNetGoogle Scholar
  61. [61]
    P. Bonami, F. Margot, Cut generation through binarization, IPCO 2014, eds. by J. Lee, J. Vygen. LNCS, vol 8494 (2014) pp. 174–185Google Scholar
  62. [62]
    J.A. Bondy, U.S.R. Murty, Graph Theory (Springer, New York, 2008)zbMATHGoogle Scholar
  63. [63]
    V. Borozan, G. Cornuéjols, Minimal valid inequalities for integer constraints. Math. Oper. Res. 34, 538–546 (2009)zbMATHMathSciNetGoogle Scholar
  64. [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)zbMATHMathSciNetGoogle Scholar
  65. [65]
    C.A. Brown, L. Finkelstein, P.W. Purdom, Backtrack Searching in the Presence of Symmetry, Nordic Journal of Computing 3, 203–219 (1996)MathSciNetGoogle Scholar
  66. [66]
    S. Burer, D. Vandenbussche, Solving lift-and-project relaxations of binary integer programs. SIAM J. Optim. 16, 726–750 (2006)zbMATHMathSciNetGoogle Scholar
  67. [67]
    A. Caprara, M. Fischetti, \(\{0, \frac{1} {2}\}\) Chvátal–Gomory cuts. Math. Program. 74, 221–235 (1996)Google Scholar
  68. [68]
    A. Caprara, A.N. Letchford, On the separation of split cuts and related inequalities. Math. Program. B 94, 279–294 (2003)zbMATHMathSciNetGoogle Scholar
  69. [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. [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. 139–169MathSciNetGoogle Scholar
  71. [71]
    M. Chudnovsky, G. Cornuéjols, X. Liu, P. Seymour, K. Vusković, Recognizing Berge graphs. Combinatorica 25, 143–186 (2005)zbMATHMathSciNetGoogle Scholar
  72. [72]
    M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)zbMATHMathSciNetGoogle Scholar
  73. [73]
    V. Chvátal, Edmonds polytopes and a hierarchy of combinatorial optimization. Discrete Math. 4, 305–337 (1973)zbMATHMathSciNetGoogle Scholar
  74. [74]
    V. Chvátal, On certain polytopes associated with graphs. J. Combin. Theory B 18, 138–154 (1975)zbMATHGoogle Scholar
  75. [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. [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. [77]
    M. Conforti, G. Cornuéjols, G. Zambelli, A geometric perspective on lifting. Oper. Res. 59, 569–577 (2011)zbMATHMathSciNetGoogle Scholar
  78. [78]
    M. Conforti, G. Cornuéjols, G. Zambelli, Equivalence between intersection cuts and the corner polyhedron. Oper. Res. Lett. 38, 153–155 (2010)zbMATHMathSciNetGoogle Scholar
  79. [79]
    M. Conforti, G. Cornuéjols, G. Zambelli, Extended formulations in combinatorial optimization. 4OR 8, 1–48 (2010)Google Scholar
  80. [80]
    M. Conforti, G. Cornuéjols, G. Zambelli, Corner polyhedron and intersection cuts. Surv. Oper. Res. Manag. Sci. 16, 105–120 (2011)Google Scholar
  81. [81]
    M. Conforti, M. Di Summa, F. Eisenbrand, L.A. Wolsey, Network formulations of mixed-integer programs. Math. Oper. Res. 34, 194–209 (2009)zbMATHMathSciNetGoogle Scholar
  82. [82]
    M. Conforti, L.A. Wolsey, Compact formulations as unions of polyhedra. Math. Program. 114, 277–289 (2008)zbMATHMathSciNetGoogle Scholar
  83. [83]
    M. Conforti, L.A. Wolsey, G. Zambelli, Split, MIR and Gomory inequalities (2012 submitted)Google Scholar
  84. [84]
    S.A. Cook, The complexity of theorem-proving procedures, in Proceedings 3rd STOC (Association for Computing Machinery, New York, 1971), pp. 151–158Google Scholar
  85. [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. 387–430Google Scholar
  86. [86]
    W.J. Cook, In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation (Princeton University Press, Princeton, 2012)Google Scholar
  87. [87]
    W.J. Cook, W.H. Cunningham, W.R. Pulleyblank, A. Schrijver, Combinatorial Optimization (Wiley, New York, 1998)zbMATHGoogle Scholar
  88. [88]
    W.J. Cook, S. Dash, R. Fukasawa, M. Goycoolea, Numerically accurate Gomory mixed-integer cuts. INFORMS J. Comput. 21, 641–649 (2009)zbMATHMathSciNetGoogle Scholar
  89. [89]
    W.J. Cook, J. Fonlupt, A. Schrijver, An integer analogue of Carathéodory’s theorem. J. Combin. Theory B 40, 63–70 (1986)zbMATHMathSciNetGoogle Scholar
  90. [90]
    W.J. Cook, R. Kannan, A. Schrijver, Chvátal closures for mixed integer programming problems. Math. Program. 47, 155–174 (1990)zbMATHMathSciNetGoogle Scholar
  91. [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)zbMATHMathSciNetGoogle Scholar
  92. [92]
    G. Cornuéjols, Combinatorial Optimization: Packing and Covering. SIAM Monograph, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 74 (2001)Google Scholar
  93. [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)zbMATHGoogle Scholar
  94. [94]
    G. Cornuéjols, Y. Li, On the rank of mixed 0,1 polyhedra. Math. Program. A 91, 391–397 (2002)zbMATHGoogle Scholar
  95. [95]
    G. Cornuéjols, Y. Li, A connection between cutting plane theory and the geometry of numbers. Math. Program. A 93, 123–127 (2002)zbMATHGoogle Scholar
  96. [96]
    G. Cornuéjols, R. Tütüncü, Optimization Methods in Finance (Cambridge University Press, Cambridge, 2007)zbMATHGoogle Scholar
  97. [97]
    A.M. Costa, A survey on Benders decomposition applied to fixed-charge network design problems. Comput. Oper. Res. 32, 1429–1450 (2005)MathSciNetGoogle Scholar
  98. [98]
    H. Crowder, M.W. Padberg, Solving large-scale symmetric travelling salesman problems to optimality. Manag. Sci. 26, 495–509 (1980)zbMATHMathSciNetGoogle Scholar
  99. [99]
    H. Crowder, E. Johnson, M.W. Padberg, Solving large scale zero-one linear programming problems. Oper. Res. 31, 803–834 (1983)zbMATHGoogle Scholar
  100. [100]
    R.J. Dakin, A tree-search algorithm for mixed integer programming problems. Comput. J. 8, 250–255 (1965)zbMATHMathSciNetGoogle Scholar
  101. [101]
    E. Danna, E. Rothberg, C. Le Pape, Exploring relaxation induced neighborhoods to improve MIP solutions. Math. Program. A 102, 71–90 (2005)zbMATHGoogle Scholar
  102. [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. 339–347Google Scholar
  103. [103]
    G. Dantzig. R. Fulkerson, S. Johnson, Solution of a large-scale traveling-salesman problem. Oper. Res. 2, 393–410 (1954)MathSciNetGoogle Scholar
  104. [104]
    G.B. Dantzig, P. Wolfe, Decomposition principle for linear programs. Oper. Res. 8, 101–111 (1960)zbMATHGoogle Scholar
  105. [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. 101–180Google Scholar
  106. [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)zbMATHMathSciNetGoogle Scholar
  107. [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. 337–351Google Scholar
  108. [108]
    R. Dechter, Constraint Processing (Morgan Kaufmann, San Francisco, 2003)Google Scholar
  109. [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)zbMATHGoogle Scholar
  110. [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. [111]
    R. de Wolf, Nondeterministic quantum query and communication complexities. SIAM J. Comput. 32, 681–699 (2003)zbMATHMathSciNetGoogle Scholar
  112. [112]
    A. Del Pia, R. Weismantel, Relaxations of mixed integer sets from lattice-free polyhedra. 4OR 10, 221–244 (2012)Google Scholar
  113. [113]
    A. Del Pia, R. Weismantel, On convergence in mixed integer programming. Math. Program. 135, 397–412 (2012)zbMATHMathSciNetGoogle Scholar
  114. [114]
    J. Desrosiers, F. Soumis, M. Desrochers, Routing with time windows by column generation. Networks 14, 545–565 (1984)zbMATHGoogle Scholar
  115. [115]
    S.S. Dey, Q. Louveaux, Split rank of triangle and quadrilateral inequalities. Math. Oper. Res. 36, 432–461 (2011)zbMATHMathSciNetGoogle Scholar
  116. [116]
    S. S. Dey, D.A. Morán, On maximal S-free convex sets. SIAM J. Discrete Math. 25(1), 379–393 (2011)zbMATHMathSciNetGoogle Scholar
  117. [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)zbMATHMathSciNetGoogle Scholar
  118. [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. 463–475MathSciNetGoogle Scholar
  119. [119]
    S.S. Dey, L.A. Wolsey, Constrained infinite group relaxations of MIPs. SIAM J. Optim. 20, 2890–2912 (2010)zbMATHMathSciNetGoogle Scholar
  120. [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. [121]
    J.-P. Doignon, Convexity in cristallographical lattices. J. Geom. 3, 71–85 (1973)zbMATHMathSciNetGoogle Scholar
  122. [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)zbMATHMathSciNetGoogle Scholar
  123. [123]
    J. Edmonds, Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)zbMATHMathSciNetGoogle Scholar
  124. [124]
    J. Edmonds, Maximum matching and a polyhedron with 0,1-vertices. J. Res. Natl. Bur. Stand. B 69, 125–130 (1965)zbMATHMathSciNetGoogle Scholar
  125. [125]
    J. Edmonds, Systems of distinct representatives and linear algebra. J. Res. Natl. Bur. Stand. B 71, 241–245 (1967)zbMATHMathSciNetGoogle Scholar
  126. [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. 69–87Google Scholar
  127. [127]
    J. Edmonds, D.R. Fulkerson, Bottleneck extrema. J. Combin. Theory 8, 299–306 (1970)zbMATHMathSciNetGoogle Scholar
  128. [128]
    J. Edmonds, R. Giles, A min-max relation for submodular functions on graphs. Ann. Discrete Math. 1, 185–204 (1977)MathSciNetGoogle Scholar
  129. [129]
    J. Edmonds, R.M. Karp, Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19, 248–264 (1972)zbMATHGoogle Scholar
  130. [130]
    F. Eisenbrand, On the membership problem for the elementary closure of a polyhedron. Combinatorica 19, 297–300 (1999)zbMATHMathSciNetGoogle Scholar
  131. [131]
    F. Eisenbrand, G. Shmonin, Carathéodory bounds on integer cones. Oper. Res. Lett. 34, 564–568 (2006)zbMATHMathSciNetGoogle Scholar
  132. [132]
    F. Eisenbrand, A.S. Schulz, Bounds on the Chvátal rank of polytopes in the 0/1 cube. Combinatorica 23, 245–261 (2003)zbMATHMathSciNetGoogle Scholar
  133. [133]
    D. Erlenkotter, A dual-based procedure for uncapacitated facility location. Oper. Res. 26, 992–1009 (1978)zbMATHMathSciNetGoogle Scholar
  134. [134]
    T. Fahle, S. Shamberger, M. Sellmann, Symmetry Breaking, CP 2001. LNCS, vol. 2239 (2001), pp. 93–107Google Scholar
  135. [135]
    Gy. Farkas, On the applications of the mechanical principle of Fourier, Mathematikai és Természettudományi Értesotö 12, 457–472 (1894)Google Scholar
  136. [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. [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)zbMATHMathSciNetGoogle Scholar
  138. [138]
    M.L. Fischer, The Lagrangian relaxation method for solving integer programming problems. Manag. Sci. 27, 1–18 (1981)Google Scholar
  139. [139]
    M. Fischetti, F. Glover, A. Lodi, The feasibility pump. Math. Program. 104, 91–104 (2005)zbMATHMathSciNetGoogle Scholar
  140. [140]
    M. Fischetti, A. Lodi, Local branching. Math. Program. B 98, 23–47 (2003)zbMATHMathSciNetGoogle Scholar
  141. [141]
    M. Fischetti, A. Lodi, Optimizing over the first Chvátal closure. Math. Program. 110, 3–20 (2007)zbMATHMathSciNetGoogle Scholar
  142. [142]
    M. Fischetti, A. Lodi, A. Tramontani, On the separation of disjunctive cuts. Math. Program. A 128, 205–230 (2011)zbMATHMathSciNetGoogle Scholar
  143. [143]
    M. Fischetti, D. Salvagnin, C. Zanette, A note on the selection of Benders’ cuts. Math. Program. B 124, 175–182 (2010)zbMATHMathSciNetGoogle Scholar
  144. [144]
    R. Fortet, Applications de l’algèbre de Boole en recherche opérationnelle. Revue Française de Recherche Opérationnelle 4, 17–26 (1960)Google Scholar
  145. [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. 317–319Google Scholar
  146. [146]
    L.R. Ford Jr., D.R. Fulkerson, Maximal flow through a network. Can. J. Math. 8, 399–404 (1956)zbMATHMathSciNetGoogle Scholar
  147. [147]
    L.R. Ford Jr., D.R. Fulkerson, Flows in Networks (Princeton University Press, Princeton, 1962)zbMATHGoogle Scholar
  148. [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. [149]
    A. Frank, E. Tardos, An application of simultaneous Diophantine approximation in combinatorial optimization. Combinatorica 7, 49–65 (1987)zbMATHMathSciNetGoogle Scholar
  150. [150]
    R. M. Freund, J.B. Orlin, On the complexity of four polyhedral set containment problems. Math. Program. 33, 139–145 (1985)zbMATHMathSciNetGoogle Scholar
  151. [151]
    A.M. Frieze, M. Jerrum, Improved approximation algorithms for MAX k-CUT and MAX BISECTION. Algorithmica 18, 67–81 (1997)zbMATHMathSciNetGoogle Scholar
  152. [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. [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. [154]
    D.R. Fulkerson, Blocking and anti-blocking pairs of polyhedra. Math. Program. 1, 168–194 (1971)zbMATHMathSciNetGoogle Scholar
  155. [155]
    D.R. Fulkerson, Anti-blocking polyhedra. J. Combin. Theory B 12, 50–71 (1972)zbMATHMathSciNetGoogle Scholar
  156. [156]
    D.R Fulkerson, Blocking polyhedra, B Harris (Ed.), Graph Theory and Its Applications, Academic Press, New York 93–112 (1970)Google Scholar
  157. [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. [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)zbMATHGoogle Scholar
  159. [159]
    R.S. Garfinkel, G. Nemhauser, Integer Programming (Wiley, New York, 1972)zbMATHGoogle Scholar
  160. [160]
    C.F. Gauss, Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium (F. Perthes & J.H. Besser, Hamburg, 1809)Google Scholar
  161. [161]
    A.M. Geoffrion, Generalized Benders decomposition. J. Optim. Theory Appl. 10, 237–260 (1972)zbMATHMathSciNetGoogle Scholar
  162. [162]
    A.M. Geoffrion, Lagrangean relaxation for integer programming. Math. Program. Study 2, 82–114 (1974)MathSciNetGoogle Scholar
  163. [163]
    A.M. Geoffrion, G.W. Graves, Multicommodity distribution design by Benders’ decomposition. Manag. Sci. 20, 822–844 (1974)zbMATHMathSciNetGoogle Scholar
  164. [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. [165]
    A. Ghouila-Houri, Caractérisation des matrices totalement unimodulaires. Comptes Rendus Hebdomadaires des Scéances de l’Académie des Sciences (Paris) 254, 1192–1194 (1962)zbMATHMathSciNetGoogle Scholar
  166. [166]
    F.R. Giles, W.R. Pulleyblank, Total dual integrality and integer polyhedra. Linear Algebra Appl. 25, 191–196 (1979)zbMATHMathSciNetGoogle Scholar
  167. [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. [168]
    P.C. Gilmore, R.E. Gomory, A linear programming approach to the cutting-stock problem. Oper. Res. 9, 849–859 (1961)zbMATHMathSciNetGoogle Scholar
  169. [169]
    M.X. Goemans, Worst-case comparison of valid inequalities for the TSP. Math. Program. 69, 335–349 (1995)zbMATHMathSciNetGoogle Scholar
  170. [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. [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. [172]
    M.X. Goemans, L. Tunçel, When does the positive semidefiniteness constraint help in lifting procedures. Math. Oper. Res. 26, 796–815 (2001)zbMATHMathSciNetGoogle Scholar
  173. [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)zbMATHMathSciNetGoogle Scholar
  174. [174]
    J.L. Goffin, Variable metric relaxation methods, part II: the ellipsoid method. Math. Program. 30, 147–162 (1984)zbMATHMathSciNetGoogle Scholar
  175. [175]
    R.E. Gomory, Outline of an algorithm for integer solutions to linear programs. Bull. Am. Math. Soc. 64, 275–278 (1958)zbMATHMathSciNetGoogle Scholar
  176. [176]
    R.E. Gomory, An algorithm for the mixed integer problem. Tech. Report RM-2597 (The Rand Corporation, 1960)Google Scholar
  177. [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. 269–302Google Scholar
  178. [178]
    R.E. Gomory, Some polyhedra related to combinatorial problems. Linear Algebra Appl. 2, 451–558 (1969)zbMATHMathSciNetGoogle Scholar
  179. [179]
    R.E. Gomory, E.L. Johnson, Some continuous functions related to corner polyhedra I. Math. Program. 3, 23–85 (1972)zbMATHMathSciNetGoogle Scholar
  180. [180]
    R.E. Gomory, E.L. Johnson, T-space and cutting planes. Math. Program. 96, 341–375 (2003)zbMATHMathSciNetGoogle Scholar
  181. [181]
    J. Gouveia, P. Parrilo, R. Thomas, Theta bodies for polynomial ideals. SIAM J. Optim. 20, 2097–2118 (2010)zbMATHMathSciNetGoogle Scholar
  182. [182]
    J. Gouveia, P. Parrilo, R. Thomas, Lifts of convex sets and cone factorizations. Math. Oper. Res. 38, 248–264 (2013)zbMATHMathSciNetGoogle Scholar
  183. [183]
    M. Grötschel, Polyedrische Charackterisierungen kombinatorischer Optimierungsprobleme (Anton Hain, Meisenheim/Glan, 1977)Google Scholar
  184. [184]
    M. Grötschel, On the symmetric travelling salesman problem: solution of a 120-city problem. Math. Program. Study 12, 61–77 (1980)zbMATHGoogle Scholar
  185. [185]
    M. Grötschel, M. Jünger, G. Reinelt, A cutting plane algorithm for the linear ordering problem. Oper. Res. 32, 1195–1220 (1984)zbMATHMathSciNetGoogle Scholar
  186. [186]
    M. Grötschel, L. Lovász, A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)zbMATHMathSciNetGoogle Scholar
  187. [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. 167–183Google Scholar
  188. [188]
    M. Grötschel, L. Lovász, A. Schrijver, Geometric Algorithms and Combinatorial Optimization (Springer, New York, 1988)zbMATHGoogle Scholar
  189. [189]
    M. Grötschel, M.W. Padberg, On the symmetric travelling salesman problem I: inequalities. Math. Program. 16, (1979) 265–280zbMATHGoogle Scholar
  190. [190]
    B. Grünbaum, Convex Polytopes (Wiley-Interscience, London, 1967)zbMATHGoogle Scholar
  191. [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)zbMATHMathSciNetGoogle Scholar
  192. [192]
    Z. Gu, G.L. Nemhauser, M.W.P. Savelsbergh, Sequence independent lifting in mixed integer programming. J. Combin. Optim. 1, 109–129 (2000)MathSciNetGoogle Scholar
  193. [193]
    C. Guéret, C. Prins, M. Servaux, Applications of Optimization with Xpress (Dash Optimization Ltd., London, 2002)Google Scholar
  194. [194]
    M. Guignard, S. Kim, Lagrangean decomposition for integer programming: theory and applications. RAIRO 21, 307–323 (1987)zbMATHMathSciNetGoogle Scholar
  195. [195]
    O. Günlük, Y. Pochet, Mixing mixed-integer inequalities. Math. Program. 90, 429–458 (2001)zbMATHMathSciNetGoogle Scholar
  196. [196]
    W. Harvey, Computing two-dimensional integer hulls. SIAM J. Comput. 28, 2285–2299 (1999)zbMATHMathSciNetGoogle Scholar
  197. [197]
    M. Held, R.M. Karp, The traveling-salesman problem and minimum spanning trees. Oper. Res. 18, 1138–1162 (1970)zbMATHMathSciNetGoogle Scholar
  198. [198]
    M. Held, R.M. Karp, The traveling-salesman problem and minimum spanning trees: part II. Math. Program. 1, 6–25 (1971)zbMATHMathSciNetGoogle Scholar
  199. [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. 247–254Google Scholar
  200. [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)zbMATHGoogle Scholar
  201. [201]
    J.-B. Hiriart-Urruty, C. Lemaréchal. Fundamentals of Convex Analysis (Springer, New York, 2001)zbMATHGoogle Scholar
  202. [202]
    D.S. Hirschberg, C.K. Wong, A polynomial algorithm for the knapsack problem in two variables. J. ACM 23, 147–154 (1976)zbMATHMathSciNetGoogle Scholar
  203. [203]
    A.J. Hoffman, A generalization of max-flow min-cut. Math. Program. 6, 352–259 (1974)zbMATHGoogle Scholar
  204. [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. 223–246Google Scholar
  205. [205]
    J.N. Hooker, Needed: an empirical science of algorithms. Oper. Res. 42, 201–212 (1994)zbMATHGoogle Scholar
  206. [206]
    J. Hooker, Integrated Methods for Optimization. International Series in Operations Research and Management Science (Springer, New York, 2010)Google Scholar
  207. [207]
    R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 2013)zbMATHGoogle Scholar
  208. [208]
    C.A.J. Hurkens, Blowing up convex sets in the plane. Linear Algebra Appl. 134, 121–128 (1990)zbMATHMathSciNetGoogle Scholar
  209. [209]
    S. Iwata, L. Fleischer, S. Fujishige, A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions. J. ACM 48, 761–777 (2001)zbMATHMathSciNetGoogle Scholar
  210. [210]
    R.G. Jeroslow, There cannot be any algorithm for integer programming with quadratic constraints. Oper. Res. 21, 221–224 (1973)zbMATHMathSciNetGoogle Scholar
  211. [211]
    R.G. Jeroslow, Representability in mixed integer programming, I: characterization results. Discrete Appl. Math. 17, 223–243 (1987)zbMATHMathSciNetGoogle Scholar
  212. [212]
    R.G Jeroslow, On defining sets of vertices of the hypercube by linear inequalities. Discrete Math. 11, 119–124 (1975)zbMATHMathSciNetGoogle Scholar
  213. [213]
    R.G Jeroslow, J.K. Lowe, Modelling with integer variables. Math. Program. Stud. 22, 167–184 (1984)zbMATHMathSciNetGoogle Scholar
  214. [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. 187–204Google Scholar
  215. [215]
    E.L. Johnson, On the group problem for mixed integer programming. Math. Program. Study 2, 137–179 (1974)Google Scholar
  216. [216]
    E.L. Johnson, Characterization of facets for multiple right-hand choice linear programs. Math. Program. Study 14, 112–142 (1981)zbMATHGoogle Scholar
  217. [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)zbMATHGoogle Scholar
  218. [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. [219]
    V. Kaibel, Extended formulations in combinatorial optimization. Optima 85, 2–7 (2011)Google Scholar
  220. [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. 287–300Google Scholar
  221. [221]
    V. Kaibel, K. Pashkovich, D.O. Theis, Symmetry matters for sizes of extended formulations. SIAM J. Discrete Math. 26(3), 1361–1382 (2012)zbMATHMathSciNetGoogle Scholar
  222. [222]
    V. Kaibel, M.E. Pfetsch, Packing and partitioning orbitopes. Math. Program. 114, 1–36 (2008)zbMATHMathSciNetGoogle Scholar
  223. [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. [224]
    V. Kaibel, S. Weltge, Lower bounds on the sizes of integer programs without additional variables. arXiv:1311.3255 (2013)Google Scholar
  225. [225]
    R. Kannan, A polynomial algorithm for the two-variable integer programming problem. J. ACM 27, 118–122 (1980)zbMATHGoogle Scholar
  226. [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. 193–206Google Scholar
  227. [227]
    R. Kannan, Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12, 415–440 (1987)zbMATHMathSciNetGoogle Scholar
  228. [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)zbMATHMathSciNetGoogle Scholar
  229. [229]
    N. Karmarkar, A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–395 (1984)zbMATHMathSciNetGoogle Scholar
  230. [230]
    D.R. Karger, Global min-cuts in RNC, and other ramifications of a simple min-cut algorithm, in Proceedings of SODA (1993), pp. 21–30Google Scholar
  231. [231]
    D.R. Karger, R. Motwani, M. Sudan, Approximate graph coloring by semidefinite programming. J. ACM 45, 246–265 (1998)zbMATHMathSciNetGoogle Scholar
  232. [232]
    R.M. Karp, Reducubility among combinatorial problems, in Complexity of Computer Computations (Plenum Press, New York, 1972), pp. 85–103Google Scholar
  233. [233]
    R.M. Karp, C.H. Papadimitriou, On linear characterizations of combinatorial optimization problems. SIAM J. Comput. 11, 620–632 (1982)zbMATHMathSciNetGoogle Scholar
  234. [234]
    H. Kellerer, U. Pferschy, D. Pisinger, Knapsack Problems (Springer, Berlin, 2004)zbMATHGoogle Scholar
  235. [235]
    L.G. Khachiyan, A polynomial algorithm in linear programming. Soviet Math. Dokl. 20, 191–194 (1979)zbMATHGoogle Scholar
  236. [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)zbMATHMathSciNetGoogle Scholar
  237. [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. [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)MathSciNetGoogle Scholar
  239. [239]
    D. Klabjan, G.L. Nemhauser, C. Tovey, The complexity of cover inequality separation. Oper. Res. Lett. 23, 35–40 (1998)zbMATHMathSciNetGoogle Scholar
  240. [240]
    V. Klee, G.J. Minty, How good is the simplex algorithm? in Inequalities, III, ed. by O. Shisha (Academic, New York, 1972), pp. 159–175Google Scholar
  241. [241]
    M. Köppe, Q. Louveaux, R. Weismantel, Intermediate integer programming representations using value disjunctions. Discrete Optim. 5, 293–313 (2008)zbMATHMathSciNetGoogle Scholar
  242. [242]
    M. Köppe, R. Weismantel, A mixed-integer Farkas lemma and some consequences. Oper. Res. Lett. 32, 207–211 (2004)zbMATHMathSciNetGoogle Scholar
  243. [243]
    B. Korte, J. Vygen, Combinatorial Optimization: Theory and Algorithms (Springer, Berlin/Hidelberg, 2000)Google Scholar
  244. [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)zbMATHMathSciNetGoogle Scholar
  245. [245]
    H.W. Kuhn, The Hungarian method for the assignment problem. Naval Res. Logistics Q. 2, 83–97 (1955)Google Scholar
  246. [246]
    A.H. Land, A.G. Doig, An automatic method of solving discrete programming problems. Econometrica 28, 497–520 (1960)zbMATHMathSciNetGoogle Scholar
  247. [247]
    J.B. Lasserre, An Explicit Exact SDP Relaxation for Nonlinear 0–1 Programs. Lecture Notes in Computer Science, vol. 2081 (2001), pp. 293–303MathSciNetGoogle Scholar
  248. [248]
    J.B. Lasserre, Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)zbMATHMathSciNetGoogle Scholar
  249. [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. [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. 393–514Google Scholar
  251. [251]
    E. L. Lawler, Covering problems: duality relations and a method of solution. SIAM J. Appl. Math. 14, 1115–1132 (1966)zbMATHMathSciNetGoogle Scholar
  252. [252]
    E. L. Lawler, Combinatorial Optimization: Networks and Matroids (Holt, Rinehart and Winston, New York, 1976)zbMATHGoogle Scholar
  253. [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)zbMATHGoogle Scholar
  254. [254]
    A. Lehman, On the width-length inequality. Math. Program. 17, 403–417 (1979)zbMATHMathSciNetGoogle Scholar
  255. [255]
    A.K. Lenstra, H.W. Lenstra, L. Lovász, Factoring polynomials with rational coefficients. Math. Ann. 261, 515–534 (1982)zbMATHMathSciNetGoogle Scholar
  256. [256]
    H.W. Lenstra, Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983)zbMATHMathSciNetGoogle Scholar
  257. [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)zbMATHMathSciNetGoogle Scholar
  258. [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. [259]
    L. Lovász, Normal hypergraphs and the perfect graph conjecture. Discrete Math. 2, 253–267 (1972)zbMATHMathSciNetGoogle Scholar
  260. [260]
    L. Lovász, On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25, 1–7 (1979)zbMATHGoogle Scholar
  261. [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. 177–201Google Scholar
  262. [262]
    L. Lovász, M.D. Plummer, Matching Theory (Akadémiai Kiadó, Budapest, 1986) [Also: North Holland Mathematics Studies, vol. 121 (North Holland, Amsterdam)]zbMATHGoogle Scholar
  263. [263]
    L. Lovász, H.E. Scarf, The generalized basis reduction algorithm. Math. Oper. Res. 17, 751–764 (1992)zbMATHMathSciNetGoogle Scholar
  264. [264]
    L. Lovász, A. Schrijver, Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1, 166–190 (1991)zbMATHMathSciNetGoogle Scholar
  265. [265]
    T.L. Magnanti, R.T. Wong, Accelerated Benders decomposition: algorithmic enhancement and model selection criteria. Oper. Res. 29, 464–484 (1981)zbMATHMathSciNetGoogle Scholar
  266. [266]
    H. Marchand, L.A. Wolsey, Aggregation and mmixed integer rounding to solve MIPs. Oper. Res. 49, 363–371 (2001)zbMATHMathSciNetGoogle Scholar
  267. [267]
    F. Margot, Pruning by isomorphism in branch-and-cut. Math. Program. 94, 71–90 (2002)zbMATHMathSciNetGoogle Scholar
  268. [268]
    S. Martello, P. Toth, Knapsack Problems: Algorithms and Computer Implementations (Wiley, Chichester, 1990)zbMATHGoogle Scholar
  269. [269]
    R.K. Martin, Generating alternative mixed integer programming models using variable definition. Oper. Res. 35, 820–831 (1987)zbMATHMathSciNetGoogle Scholar
  270. [270]
    R.K. Martin, Using separation algorithms to generate mixed integer model reformulations. Oper. Res. Lett. 10(3), 119–128 (1991)zbMATHMathSciNetGoogle Scholar
  271. [271]
    R.K. Martin, R.L. Rardin, B.A. Campbell, Polyhedral characterization of discrete dynamic programming. Oper. Res. 38, 127–138 (1990)zbMATHMathSciNetGoogle Scholar
  272. [272]
    J.F. Maurras, Bon algorithmes, vieilles idées, Note E.d.F. HR 32.0320 (1978)Google Scholar
  273. [273]
    J.F. Maurras, K. Truemper, M. Agkül, Polynomial algorithms for a class of linear programs. Math. Program. 21, 121–136 (1981)zbMATHGoogle Scholar
  274. [274]
    C.C. McGeogh, Experimental analysis of algorithms. Notices Am. Math. Assoc. 48, 204–311 (2001)Google Scholar
  275. [275]
    B.D. McKay, Practical graph isomorphism. Congressus Numerantium 30, 45–87 (1981)MathSciNetGoogle Scholar
  276. [276]
    R.R. Meyer, On the existence of optimal solutions to integer and mixed integer programming problems. Math. Program. 7, 223–235 (1974)zbMATHGoogle Scholar
  277. [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. 92–98Google Scholar
  278. [278]
    C.E. Miller, A.W. Tucker, R.A. Zemlin, Integer programming formulation of traveling salesman problems. J. ACM 7, 326–329 (1960)zbMATHMathSciNetGoogle Scholar
  279. [279]
    H. Minkowski, Geometrie der Zahlen (Erste Lieferung) (Teubner, Leipzig, 1896)Google Scholar
  280. [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. [281]
    J. Munkres, Algorithms for the assignment and transportation problems. J. SIAM 5, 32–38 (1957)zbMATHMathSciNetGoogle Scholar
  282. [282]
    H. Nagamochi, T. Ibaraki, Computing edge-connectivity in multiple and capacitated graphs. SIAM J. Discrete Math. 5, 54–66 (1992)zbMATHMathSciNetGoogle Scholar
  283. [283]
    G.L. Nemhauser, L.E. Trotter Jr., Properties of vertex packing and independence system polyhedra. Math. Program. 6, 48–61 (1974)zbMATHMathSciNetGoogle Scholar
  284. [284]
    G.L. Nemhauser, L.E. Trotter Jr., Vertex packings: structural properties and algorithms. Math. Program. 8, 232–248 (1975)zbMATHMathSciNetGoogle Scholar
  285. [285]
    G.L. Nemhauser, L.A. Wolsey, Integer and Combinatorial Optimization (Wiley, New York, 1988)zbMATHGoogle Scholar
  286. [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)zbMATHMathSciNetGoogle Scholar
  287. [287]
    Y.E. Nesterov, Smooth minimization of non-smooth functions. Math. Program. A 103, 127–152 (2005)zbMATHMathSciNetGoogle Scholar
  288. [288]
    Y.E. Nesterov, Semidefinite relaxation and nonconvex quadratic optimization. Optim. Methods Softw. 12, 1–20 (1997)MathSciNetGoogle Scholar
  289. [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. [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. [291]
    Y.E. Nesterov, A.S. Nemirovski, Interior Point Polynomial Algorithms in Convex Programming (SIAM, Philadelphia, 1994)zbMATHGoogle Scholar
  292. [292]
    J. Ostrowski, J.T. Linderoth, F. Rossi, S. Smriglio, Solving large Steiner triple covering problems. Oper. Res. Lett. 39, 127–131 (2011)zbMATHMathSciNetGoogle Scholar
  293. [293]
    J. Ostrowski, J. Linderoth, F. Rossi, S. Smriglio, Orbital branching. Math. Program. 126, 147–178 (2011)zbMATHMathSciNetGoogle Scholar
  294. [294]
    J.H. Owen, S. Mehrotra, A disjunctive cutting plane procedure for general mixed-integer linear programs. Math. Program. A 89, 437–448 (2001)zbMATHMathSciNetGoogle Scholar
  295. [295]
    J.H. Owen, S. Mehrotra, On the value of binary expansions for general mixed-integer linear programs. Oper. Res. 50, 810–819 (2002)zbMATHMathSciNetGoogle Scholar
  296. [296]
    J. Oxley, Matroid Theory (Oxford University Press, New York, 2011)zbMATHGoogle Scholar
  297. [297]
    M.W. Padberg, On the facial structure of set packing polyhedra. Math. Program. 5, 199–215 (1973)zbMATHMathSciNetGoogle Scholar
  298. [298]
    M.W. Padberg, A note on zero-one programming. Oper. Res. 23, 833–837 (1975)zbMATHMathSciNetGoogle Scholar
  299. [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. [300]
    M.W. Padberg, M.R. Rao, Odd minimum cut-sets and b-matchings. Math. Oper. Res. 7, 67–80 (1982)zbMATHMathSciNetGoogle Scholar
  301. [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)zbMATHMathSciNetGoogle Scholar
  302. [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)zbMATHMathSciNetGoogle Scholar
  303. [303]
    M. Padberg, T.J. Van Roy, L.A. Wolsey, Valid linear inequalities for fixed charge problems. Oper. Res. 33, 842–861 (1985)zbMATHMathSciNetGoogle Scholar
  304. [304]
    J. Pap, Recognizing conic TDI systems is hard. Math. Program. 128, 43–48 (2011)zbMATHMathSciNetGoogle Scholar
  305. [305]
    C.H. Papadimitriou, On the complexity of integer programming. J. ACM 28, 765–768 (1981)zbMATHMathSciNetGoogle Scholar
  306. [306]
    J. Patel, J.W. Chinneck, Active-constraint variable ordering for faster feasibility of mixed integer linear programs. Math. Program. 110, 445–474 (2007)zbMATHMathSciNetGoogle Scholar
  307. [307]
    J. Petersen, Die Theorie der regulären graphs. Acta Matematica 15, 193–220 (1891)zbMATHGoogle Scholar
  308. [308]
    Y. Pochet, L.A. Wolsey, Polyhedra for lot-sizing with Wagner–Whitin costs. Math. Program. 67, 297–324 (1994)zbMATHMathSciNetGoogle Scholar
  309. [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. [310]
    B.T. Poljak, A general method for solving extremum problems. Soviet Math. Dokl. 8, 593–597 (1967)Google Scholar
  311. [311]
    C.H. Papadimitriou, M. Yannakakis, On recognizing integer polyhedra. Combinatorica 10, 107–109 (1990)zbMATHMathSciNetGoogle Scholar
  312. [312]
    M. Queyranne, A.S. Schulz, Polyhedral approaches to machine scheduling. Preprint (1994)Google Scholar
  313. [313]
    A. Razborov, On the distributional complexity of disjointness. Theor. Comput. Sci. 106(2), 385–390 (1992)zbMATHMathSciNetGoogle Scholar
  314. [314]
    J. Renegar, A polynomial-time algorithm based on Newton’s method for linear programming. Math. Program. 40, 59–93 (1988)zbMATHMathSciNetGoogle Scholar
  315. [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. 727–801Google Scholar
  316. [316]
    R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1969)Google Scholar
  317. [317]
    T. Rothvoß, Some 0/1 polytopes need exponential size extended formulations. Math. Program. A 142, 255–268 (2012)Google Scholar
  318. [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. 263–272Google Scholar
  319. [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. [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. [321]
    M.P.W. Savelsbergh, Preprocessing and probing techniques for mixed integer programming problems. ORSA J. Comput. 6, 445–454 (1994)zbMATHMathSciNetGoogle Scholar
  322. [322]
    H.E. Scarf, An observation on the structure of production sets with indivisibilities. Proc. Natl. Acad. Sci. USA 74, 3637–3641 (1977)zbMATHMathSciNetGoogle Scholar
  323. [323]
    A. Schrijver, On cutting planes. Ann. Discrete Math. 9, 291–296 (1980)zbMATHMathSciNetGoogle Scholar
  324. [324]
    A. Schrijver, On total dual integrality. Linear Algebra Appl. 38, 27–32 (1981)zbMATHMathSciNetGoogle Scholar
  325. [325]
    A. Schrijver, Theory of Linear and Integer Programming (Wiley, New York, 1986)zbMATHGoogle Scholar
  326. [326]
    A. Schrijver, A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Combin. Theory Ser. B 80, 346–355 (2000)zbMATHMathSciNetGoogle Scholar
  327. [327]
    A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency (Springer, Berlin, 2003)Google Scholar
  328. [328]
    Á. Seress, Permutation Group Algorithms, Cambridge Tracts in Mathematics, vol. 152 (Cambridge University Press, Cambridge, 2003)Google Scholar
  329. [329]
    P.D. Seymour, Decomposition of regular matroids. J. Combin. Theory B 28, 305–359 (1980)zbMATHMathSciNetGoogle Scholar
  330. [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)MathSciNetGoogle Scholar
  331. [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. [332]
    N. Z. Shor, Cut-off method with space extension in convex programming problems. Cybernetics 13, 94–96 (1977)Google Scholar
  333. [333]
    M. Stoer, F. Wagner, A simple min-cut algorithm. J. ACM 44, 585–591 (1997)zbMATHMathSciNetGoogle Scholar
  334. [334]
    E. Tardos, A strongly polynomial algorithm to solve combinatorial linear programs. Oper. Res. 34, 250–256 (1986)zbMATHMathSciNetGoogle Scholar
  335. [335]
    R.E. Tarjan, Depth-first search and linear graph algorithms. SIAM J. Comput. 1, 146–160 (1972)zbMATHMathSciNetGoogle Scholar
  336. [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)zbMATHMathSciNetGoogle Scholar
  337. [337]
    P. Toth, D. Vigo, The Vehicle Routing Problem. Monographs on Discrete Mathematics and Applications (SIAM, Philadelphia, 2001)Google Scholar
  338. [338]
    K. Truemper, Matroid Decomposition (Academic, Boston, 1992)zbMATHGoogle Scholar
  339. [339]
    W.T. Tutte, A homotopy theorem for matroids I, II. Trans. Am. Math. Soc. 88, 905–917 (1958)MathSciNetGoogle Scholar
  340. [340]
    T.J. Van Roy, L.A. Wolsey, Solving mixed integer programming problems using automatic reformulation. Oper. Res. 35, 45–57 (1987)zbMATHMathSciNetGoogle Scholar
  341. [341]
    M. Van Vyve, The continuous mixing polyhedron. Math. Oper. Res. 30, 441–452 (2005)zbMATHMathSciNetGoogle Scholar
  342. [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. 431–502Google Scholar
  343. [343]
    R.J. Vanderbei, Linear Programming: Foundations and Extentions, 3rd edn. (Springer, New York, 2008)Google Scholar
  344. [344]
    S. Vavasis, On the complexity of nonnegative matrix factorization. SIAM J. Optim. 20, 1364–1377 (2009)zbMATHMathSciNetGoogle Scholar
  345. [345]
    V.V. Vazirani, Approximation Algorithms (Springer, Berlin, 2003)Google Scholar
  346. [346]
    J.P. Vielma, A constructive characterization of the split closure of a mixed integer linear program. Oper. Res. Lett. 35, 29–35 (2007)zbMATHMathSciNetGoogle Scholar
  347. [347]
    J.P. Vielma, Mixed integer linear programming formulation techniques to appear in SIAM Review (2014)Google Scholar
  348. [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. 3–18Google Scholar
  349. [349]
    D.P. Williamson, D.B. Shmoys, The Design of Approxiamtion Algorithms (Cambridge University Press, Cambridge, 2011)Google Scholar
  350. [350]
    L.A. Wolsey, Further facet generating procedures for vertex packing polytopes. Math. Program. 11, 158–163 (1976)zbMATHMathSciNetGoogle Scholar
  351. [351]
    L.A. Wolsey, Valid inequalities and superadditivity for 0–1 integer programs. Math. Oper. Res. 2, 66–77 (1977)zbMATHMathSciNetGoogle Scholar
  352. [352]
    L.A. Wolsey, Heuristic analysis, linear programming, and branch and bound. Math. Program. Stud. 13, 121–134 (1980)zbMATHMathSciNetGoogle Scholar
  353. [353]
    L.A. Wolsey, Integer Programming (Wiley, New York, 1999)Google Scholar
  354. [354]
    R.T. Wong, Dual ascent approach for Steiner tree problems on directed graphs. Math. Program. 28, 271–287 (1984)zbMATHGoogle Scholar
  355. [355]
    M. Yannakakis, Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43, 441–466 (1991)zbMATHMathSciNetGoogle Scholar
  356. [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. [357]
    G.M. Ziegler, Lectures on Polytopes (Springer, New York, 1995)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Michele Conforti
    • 1
  • Gérard Cornuéjols
    • 2
  • Giacomo Zambelli
    • 3
  1. 1.Department of MathematicsUniversity of PadovaPadovaItaly
  2. 2.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA
  3. 3.Department of ManagementLondon School of Economics and Political ScienceLondonUK

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