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The Group-Theoretic Approach in Mixed Integer Programming

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50 Years of Integer Programming 1958-2008

Abstract

In this chapter, we provide an overview of the mathematical foundations and recent theoretical and computational advances in the study of the grouptheoretic approach in mixed integer programming. We motivate the definition of group relaxation geometrically and present methods to optimize linear functions over this set. We then discuss fundamental results about the structure of group relaxations. We describe a variety of recent methods to derive valid inequalities for master group relaxations and review general proof techniques to show that candidate inequalities are strong (extreme) for these sets. We conclude by discussing the insights gained from computational studies aimed at gauging the strength of grouptheoretic relaxations and cutting planes for mixed integer programs.

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References

  1. A. Atamtürk, On the facets of the mixed-integer knapsack polyhedron, Mathematical Programming 98 (2003) 145–175.

    Article  MathSciNet  MATH  Google Scholar 

  2. T. Achterberg, T. Koch, and A. Martin, Branching rules revisited, Operation Research Letters 33 (2005) 42–54.

    Article  MathSciNet  MATH  Google Scholar 

  3. R.K. Ahuja, T.L. Magnanti, and J.B. Orlin, Network flows: Theory, algorithms, and applications, Prentice Hall, 1993.

    Google Scholar 

  4. K. Andersen, G. Cornuéjols, and Y. Li, Reduce-and-split cuts: Improving the performance of mixed integer Gomory cuts, Management Science 51 (2005) 1720–1732.

    Article  MATH  Google Scholar 

  5. K. Andersen, Q. Louveaux, and R. Weismantel, Geometric study of mixed-integer sets from 2 rows of 2 adjacent simplex bases, Manuscript, 2009.

    Google Scholar 

  6. K. Andersen, Q. Louveaux, R. Weismantel, and L.A. Wolsey, Cutting planes from two rows of a simplex tableau, Proceedings 12th Conference on Integer and Combinatorial Optimization (M. Fischetti and D. P. Williamson, eds.), Springer-Verlag, 2007, pp. 30–42.

    Google Scholar 

  7. J. Aráoz, Polyhedral neopolarities, Ph.D. thesis, University of Waterloo, Department of Computer Sciences, Waterloo, Canada, 1974.

    Google Scholar 

  8. J. Aráoz, L. Evans, R.E. Gomory, and E.L. Johnson, Cyclic groups and knapsack facets, Mathematical Programming 96 (2003) 377–408.

    Article  MathSciNet  MATH  Google Scholar 

  9. E. Balas, A note on the group-theoretic approach to integer programming and the 0-1 case, Operations Research 21 (1973) 321–322.

    Article  MathSciNet  MATH  Google Scholar 

  10. E. Balas, S. Ceria, G. Cornuéjols, and N. Natraj, Gomory cuts revisited, Operations Research Letters 19 (1996) 1–9.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Basu, M. Conforti, G. Cornuéjols, and G. Zambelli, Minimal inequalities for an infinite relaxation of integer programs., Manuscript, 2009.

    Google Scholar 

  12. D.E. Bell, A theorem concerning the integer lattice, Studies in Applied Mathematics 56 (1977) 187–188.

    Article  MathSciNet  MATH  Google Scholar 

  13. D.E. Bell and J.F. Shapiro, A convergent duality theory for integer programming, Operations Research 25 (1977) 419–434.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Ben-Israel and A. Charnes, On some problems of diophantine programming, Cahiers du Centre d’Études de Recherche Opérationelle 4 (1962) 215–280.

    MathSciNet  MATH  Google Scholar 

  15. R.E. Bixby, E.A. Boyd, and R.R. Indovina, MIPLIB: A test set of mixed integer programming problems, SIAM News 25:2 (1992).

    Google Scholar 

  16. R.E. Bixby and E.E. Rothberg, Progress in computational mixed integer programming - A look back from the other side of the tipping point, Annals of Operations Research 149 (2007) 37–41.

    Article  MathSciNet  MATH  Google Scholar 

  17. V. Borozan and G. Cornuéjols, Minimal inequalities for integer constraints, http://integer.tepper.cmu.edu, 2007.

  18. V.J. Bowman and G.L. Nemhauser, A finiteness proof for modified Dantzig cuts in integer programming, Naval Research Logistics Quarterly 17 (1970) 309–313.

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Charnes and W.W. Cooper, Management models and industrial applications of linear programming, ii, Wiley, New York, 1961.

    MATH  Google Scholar 

  20. D.-S. Chen and S. Zionts, Comparison of some algorithms for solving the group theoretic integer programming problem, Operations Research 24 (1976) 1120–1128.

    Article  MATH  Google Scholar 

  21. V. Chvátal, Edmonds Polytopes and a Hierarchy of Combinatorial Problems, Discrete Mathematics 4 (1973) 305–337.

    Article  MathSciNet  MATH  Google Scholar 

  22. V. Chvátal, Linear programming, W. H. Freeman and Company, New York, NY, 1983.

    MATH  Google Scholar 

  23. W. Cook, S. Dash, R. Fukasawa, and M. Goycoolea, Numerically accurate Gomory mixedinteger cuts, INFORMS Journal of Computing (To appear).

    Google Scholar 

  24. G. Cornuéjols, Revival of the Gomory cuts in the 1990’s, Annals of Operations Research 149 (2007) 63–66.

    Article  MathSciNet  MATH  Google Scholar 

  25. G. Cornuéjols and Y. Li, On the rank of mixed 0-1 polyhedra, Mathematical Programming 91 (2002) 391–397.

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Cornuéjols, Y. Li, and D. Vandenbussche, K-cuts: a variation of Gomory mixed integer cuts from the LP tableau, INFORMS Journal of Computing 15 (2003) 385–396.

    Article  MathSciNet  MATH  Google Scholar 

  27. G. Cornuéjols and F. Margot, On the facets of mixed integer programs with two integer variables and two constraints, Mathematical Programming 120 (2009) 429–456.

    Article  MathSciNet  MATH  Google Scholar 

  28. G.B. Dantzig, Discrete-variable extremum problems, Operations Research 5 (1957) 266–277.

    Article  MathSciNet  MATH  Google Scholar 

  29. G.B. Dantzig, Note on solving linear programs in integers, Naval Research Logistics Quarterly 6 (1959) 75–76.

    Article  MathSciNet  Google Scholar 

  30. S. Dash, R. Fukasawa, and O. Günlük, On a generalization of the master cyclic group polyhedron, Proceedings 12th Conference on Integer and Combinatorial Optimization (M. Fischetti and D. P. Williamson, eds.), Springer-Verlag, 2007, pp. 197–209.

    Google Scholar 

  31. S. Dash, M. Goycoolea, and O. Günlük, Two step MIR inequalities for mixed integer programs., http://www.optimization-online.org/DBHTML/2006/07/1422.html, 2006.

  32. S. Dash and O. Günlük, Valid inequalities based on simple mixed-integer sets., Proceedings 10th Conference on Integer Programming and Combinatorial Optimization (D. Bienstock and G. Nemhauser, eds.), Springer-Verlag, 2004, pp. 33–45.

    Google Scholar 

  33. S. Dash and O. Günlük, Valid inequalities based on simple mixed integer set, Mathematical Programming 106 (2006) 29–53.

    Article  MathSciNet  MATH  Google Scholar 

  34. S. Dash and O. Günlük, Valid inequalities based on the interpolation procedure, Mathematical Programming 106 (2006) 111–136.

    Article  MathSciNet  MATH  Google Scholar 

  35. S. Dash and O. Günlük, On the strength of Gomory mixed integer cuts as group cuts, Mathematical Programming 115 (2008) 387–407.

    Article  MathSciNet  MATH  Google Scholar 

  36. S.S. Dey, Strong cutting planes for unstructured mixed integer programs using multiple constraints, Ph.D. thesis, Purdue University, West Lafayette, IN, USA, 2007.

    Google Scholar 

  37. S.S. Dey and J.-P.P. Richard, Sequential-merge facets for two-dimensional group problems, Proceedings 12th Conference on Integer and Combinatorial Optimization (M. Fischetti and D. P. Williamson, eds.), Springer-Verlag, 2007, pp. 30–42.

    Google Scholar 

  38. S.S. Dey and J.-P.P. Richard, Facets of the two-dimensional infinite group problems, Mathematics of Operations Research 33 (2008) 140–166.

    Article  MathSciNet  MATH  Google Scholar 

  39. S.S. Dey and J.-P.P. Richard, Some relations between facets of low- and high-dimensional group problems, Mathematical Programming (To appear).

    Google Scholar 

  40. S.S. Dey, J.-P.P. Richard, Y. Li, and L.A. Miller, Extreme inequalities for infinite group problems., Mathematical Programming 121 (2010) 145–170.

    Article  MathSciNet  MATH  Google Scholar 

  41. S.S. Dey and L.A.Wolsey, Lifting integer variables in minimal inequalities corresponding to lattice-free triangles, Proceedings 13th Conference on Integer and Combinatorial Optimization (A. Lodi, A. Panconesi, and G. Rinaldi, eds.), Springer-Verlag, 2008, pp. 463–475.

    Google Scholar 

  42. S.S. Dey and L.A. Wolsey, Two row mixed integer cuts via lifting, Tech. Report CORE DP 30, Université catholique de Louvain, Louvain-la-Neuve, Belgium, 2008.

    Google Scholar 

  43. S.S. Dey and L.A. Wolsey, Constrained infinite group relaxations of MIPs, Tech. Report CORE DP 33, Université catholique de Louvain, Louvain-la-Neuve, Belgium, 2009.

    Google Scholar 

  44. J.P. Doignon, Convexity in crystallographic lattices, Journal of Geometry 3 (1973) 71–85.

    Article  MathSciNet  MATH  Google Scholar 

  45. D. Espinoza, Computing with multiple-row Gomory cuts, Proceedings 13th Conference on Integer Programming and Combinatorial Optimization (A. Lodi, A. Panconesi, and G. Rinaldi, eds.), Springer-Verlag, 2008, pp. 214–224.

    Google Scholar 

  46. L.A. Evans, Cyclic group and knapsack facets with applications to cutting planes, Ph.D. thesis, Georgia Institute of Technology, Atlanta, GA, USA, 2002.

    Google Scholar 

  47. M. Fischetti, F. Glover, and A. Lodi, The feasibility pump, Mathematical Programming 104 (2005) 91–104.

    Article  MathSciNet  MATH  Google Scholar 

  48. M. Fischetti and A. Lodi, Local branching, Mathematical Programming 98 (2003) 23–47.

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Fischetti and M. Monaci, How tight is the corner relaxation?, Discrete Optimization 5 (2007) 262–269.

    Article  MathSciNet  MATH  Google Scholar 

  50. M. Fischetti and C. Saturni, Mixed integer cuts from cyclic groups, Mathematical Programming 109 (2007) 27–53.

    Article  MathSciNet  MATH  Google Scholar 

  51. R. Fukasawa and O. Günlük, Strengthening lattice-free cuts using non-negativity, http://www.optimization-online.org/DBHTML/2009/05/2296.html, 2009.

  52. F. Glover, Integer programming over a finite additive group, SIAM Journal on Control 7 (1969) 213–231.

    Article  MathSciNet  MATH  Google Scholar 

  53. M.X. Goemans, Worst-case comparison of valid inequalities for the TSP, Mathematical Programming 69 (1995) 335–349.

    MathSciNet  MATH  Google Scholar 

  54. R.E. Gomory, Outline of an algorithm for integer solutions to linear programs, Bulletin of the American Mathematical Society 64 (1958) 275–278.

    Article  MathSciNet  MATH  Google Scholar 

  55. R.E. Gomory, An algorithm for the mixed integer problem, Tech. Report RM-2597, RAND Corporation, 1960.

    Google Scholar 

  56. R.E. Gomory, Some relation between integer and non-integer solutions of linear program, Proceedings of National Academy of Science 53 (1965) 250–265.

    Article  MathSciNet  Google Scholar 

  57. R.E. Gomory, Some polyhedra related to combinatorial problems, Linear Algebra and its Application 2 (1969) 341–375.

    Article  MathSciNet  MATH  Google Scholar 

  58. R.E. Gomory and A.J. Hoffman, On the convergence of an integer programming process, Naval Research Logistics Quarterly 10 (1963) 121–124.

    Article  MathSciNet  MATH  Google Scholar 

  59. R.E. Gomory and E.L. Johnson, Some continuous functions related to corner polyhedra, part I, Mathematical Programming 3 (1972) 23–85.

    Article  MathSciNet  MATH  Google Scholar 

  60. R.E. Gomory and E.L. Johnson, Some continuous functions related to corner polyhedra, part II, Mathematical Programming 3 (1972) 359–389.

    Article  MathSciNet  MATH  Google Scholar 

  61. R.E. Gomory and E.L. Johnson, T-space and cutting planes, Mathematical Programming 96 (2003) 341–375.

    Article  MathSciNet  MATH  Google Scholar 

  62. R.E. Gomory, E.L. Johnson, and L. Evans, Corner polyhedra and their connection with cutting planes, Mathematical Programming 96 (2003) 321–339.

    Article  MathSciNet  MATH  Google Scholar 

  63. G.A. Gorry, W.D. Northup, and J.F. Shapiro, Computational experience with a group theoretic integer programming algorithm, Mathematical Programming 4 (1973) 171–192.

    Article  MathSciNet  MATH  Google Scholar 

  64. J. Hadamard, Résolution d’une question relative aux déterminants, Bulletin des Sciences Mathématiques 17 (1893) 30–31.

    MATH  Google Scholar 

  65. S. Hoşten and R.R. Thomas, Standard pairs and group relaxations in integer programming, Journal of Pure and Applied Algebra 139 (1999) 133–157.

    Article  MathSciNet  MATH  Google Scholar 

  66. S. Hoşten and R.R. Thomas, Gomory integer programs, Mathematical Programming 96 (2003) 271–292.

    Article  MathSciNet  MATH  Google Scholar 

  67. T.C. Hu, Integer programming and network flows, Addison-Wesley, Reading, MA, 1969.

    MATH  Google Scholar 

  68. T.C. Hu, On the asymptotic integer algorithm, Linear Algebra and its Applications 3 (1970) 279–294.

    Article  MathSciNet  MATH  Google Scholar 

  69. B. Hunsaker, Measuring facets of polyhedra to predict usefulness in branch-and-cut algorithms, Ph.D. thesis, Georgia Institute of Technology, Atlanta, USA, 2003.

    Google Scholar 

  70. R.G. Jeroslow and K.O. Kortanek, On an algorithm of Gomory, SIAM Journal on Applied Mathematics 21 (1971) 55–60.

    Article  MathSciNet  MATH  Google Scholar 

  71. E.L. Johnson, On the group problem for mixed integer programming, Mathematical Programming Study 2 (1974) 137–179.

    Article  MathSciNet  Google Scholar 

  72. E.L. Johnson, Characterization of facets for multiple right-hand side choice linear programs, Mathematical Programming Study 14 (1981) 112–142.

    Article  MATH  Google Scholar 

  73. R. Kannan and R. Bachem, Polynomial time algorithms for computing Smith and Hermite normal forms of an integer matrix, SIAM Journal on Computation 8 (1979) 499–507.

    Article  MATH  Google Scholar 

  74. K. Kianfar and Y. Fathi, Generalized mixed integer rounding valid inequalities: Facets for infinite group polyhedra, Mathematical Programming 120 (2009) 313–346.

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  76. H.W. Kuhn, On the origin of the Hungarian method, History of Mathematical Programming: A collection of personal Reminiscences (J.K. Lenstra, K. Rinnooy, and A.H.G. Schrijver, eds.), Elsevier Science Publisher, 1991, pp. 77–81.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  78. A.N. Letchford and A. Lodi, Strengthening Chvátal-Gomory cuts and Gomory fractional cuts, Operations Research Letters 30 (2002) 74–82.

    Article  MathSciNet  MATH  Google Scholar 

  79. L. Lovász, Geometry of numbers and integer programming, Mathematical Programming: Recent Developments and Applications (1989) 177–210.

    Google Scholar 

  80. H. Marchand, A. Martin, R. Weismantel, and L.A. Wolsey, Cutting planes in integer and mixed integer programming, Discrete Applied Mathematics 123 (2002) 397–446.

    Article  MathSciNet  MATH  Google Scholar 

  81. H. Marchand and L.A. Wolsey, Aggregation and mixed integer rounding to solve MIPs, Operations Research 49 (2001) 363–371.

    Article  MathSciNet  MATH  Google Scholar 

  82. L.A. Miller, Y. Li, and J.-P.P. Richard, New facets for finite and infinite group problems from approximate lifting, Naval Research Logistics 55 (2008) 172–191.

    Article  MathSciNet  MATH  Google Scholar 

  83. G.L. Nemhauser and L.A. Wolsey, Integer and combinatorial optimization, Wiley-Interscience, New York, NY, 1988.

    Book  MATH  Google Scholar 

  84. F.J. Nourie and E.R. Venta, An upper bound on the number of cuts needed in Gomory’s method of integer forms, Operations Research Letters 1 (1982) 129–133.

    Article  MathSciNet  MATH  Google Scholar 

  85. Y. Pochet and L.A. Wolsey, Integer knapsacks and flow covers with divisible coefficients: polyhedra, optimization and separation, Discrete Applied Mathematics 59 (1995) 57–74.

    Article  MathSciNet  MATH  Google Scholar 

  86. J.-P.P. Richard, Y. Li, and L.A. Miller, Valid inequalities for MIPs and group polyhedra from approximate liftings, Mathematical Programming 118 (2009) 253–277.

    Article  MathSciNet  MATH  Google Scholar 

  87. D.S. Rubin and R.L. Graves, Strengthened Dantzig cuts for integer programming, Operations Research 20 (1972) 178–182.

    Article  MathSciNet  MATH  Google Scholar 

  88. H.M. Salkin and S. Morito, Integer programming by group theory: Some computational results, Tech. report, Defense Technical Information Center OAI-PMH Repository [http://stinet.dtic.mil/oai/oai] (United States), 1975.

  89. H.E. Scarf, An observation on the structure of production sets with indivisibilities, Proceedings of the National Academy of Sciences USA 74 (1977) 3637–3641.

    Article  MathSciNet  MATH  Google Scholar 

  90. A. Schrijver, Theory of linear and integer programming, John Wiley & Sons, Chichester, 1986.

    Google Scholar 

  91. J.F. Shapiro, Dynamic programming algorithms for the integer programming problem - i: The integer programming problem viewed as a knapsack type problem, Operations Research 16 (1968) 103–121.

    Article  MathSciNet  MATH  Google Scholar 

  92. J.F. Shapiro, Group theoretic algorithms for the integer programming problem - ii: Extension to a general algorithm, Operations Research 16 (1968) 928–947.

    Article  MathSciNet  MATH  Google Scholar 

  93. H.J.S. Smith, On systems of indeterminate equations and congruences, Philosophical Transactions of the Royal Society of London (A) 151 (1861) 293–326.

    Article  Google Scholar 

  94. B. Sturmfels, R. Weismantel, and G. Ziegler, Gröbner bases of lattices, corner polyhedra and integer programming, Beiträge zur Algebra und Geometrie 36 (1995) 281–298.

    MathSciNet  MATH  Google Scholar 

  95. R.R. Thomas, The structure of group relaxations, Handbooks in Operations Research and Management Science (R.Weismantel K. Aardal, G. Nemhauser, ed.), Elsevier, 2005, pp. 19– 49.

    Google Scholar 

  96. R.J. Vanderbei, Linear programming: Foundations and extensions, Kluwer Academic Publishers, 2001.

    Google Scholar 

  97. L.A. Wolsey, Extensions of the group theoretic approach in integer programming, Management Science 18 (1971) 74–83.

    Article  MathSciNet  MATH  Google Scholar 

  98. L.A.Wolsey, Group theoretic results in mixed integer programming, Operations Research 19 (1971) 1691–1697.

    Article  MathSciNet  MATH  Google Scholar 

  99. L.A. Wolsey, Generalized dynamic programming methods in integer programming, Mathematical Programming 4 (1973) 222–232.

    Article  MathSciNet  MATH  Google Scholar 

  100. L.A.Wolsey, The b-hull of an integer program, Discrete Applied Mathematics 3 (1981) 193–201.

    Article  MathSciNet  MATH  Google Scholar 

  101. G. Zambelli, On degenerate multi-row Gomory cuts, Operations Research Letters 37 (2009) 21–22.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    Jean-Philippe P. Richard

  2. Center for Operations Research and Econometrics, Université Catholique de Louvain, Louvain-la-Neuve, Belgium

    Santanu S. Dey

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  1. Jean-Philippe P. Richard
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Correspondence to Jean-Philippe P. Richard .

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Editors and Affiliations

  1. Inst. Informatik, Universität Köln, Pohligstr. 1, Köln, 50969, Germany

    Michael Jünger

  2. Fac. Sciences de Base (FSB), Ecole Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland

    Thomas M. Liebling

  3. Ensimag, Institut Polytechnique de Grenoble, avenue Félix Viallet 46, Grenoble CX 1, 38031, France

    Denis Naddef

  4. School of Industrial &, Georgia Institute of Technology, Ferst Drive NW., 765, Atlanta, 30332-0205, USA

    George L. Nemhauser

  5. IBM Corporation, Route 100 294, Somers, 10589, USA

    William R. Pulleyblank

  6. Inst. Informatik, Universität Heidelberg, Im Neuenheimer Feld 326, Heidelberg, 69120, Germany

    Gerhard Reinelt

  7. ed Informatica, CNR - Ist. Analisi dei Sistemi, Viale Manzoni 30, Roma, 00185, Italy

    Giovanni Rinaldi

  8. Center for Operations Reserach &, Université Catholique de Louvain, voie du Roman Pays 34, Leuven, 1348, Belgium

    Laurence A. Wolsey

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Richard, JP.P., Dey, S.S. (2010). The Group-Theoretic Approach in Mixed Integer Programming. In: Jünger, M., et al. 50 Years of Integer Programming 1958-2008. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68279-0_19

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