Minimal cut-generating functions are nearly extreme

  • Amitabh Basu
  • Robert Hildebrand
  • Marco Molinaro
Full Length Paper Series B


We study continuous (strongly) minimal cut generating functions for the model where all variables are integer. We consider both the original Gomory–Johnson setting as well as a recent extension by Yıldız and Cornuéjols (Math Oper Res 41:1381–1403, 2016). We show that for any continuous minimal or strongly minimal cut generating function, there exists an extreme cut generating function that approximates the (strongly) minimal function as closely as desired. In other words, the extreme functions are “dense” in the set of continuous (strongly) minimal functions.

Mathematics Subject Classification

90C10 90C57 


  1. 1.
    Balas, E., Qualizza, A.: Monoidal cut strengthening revisited. Discrete Optim. 9(1), 40–49 (2012)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Basu, A., Conforti, M., Cornuéjols, G., Zambelli, G.: Maximal lattice-free convex sets in linear subspaces. Math. Oper. Res. 35, 704–720 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Basu, A., Conforti, M., Di Summa, M.: A geometric approach to cut-generating functions. Math. Program. 151(1), 153–189 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Basu, A., Hildebrand, R., Köppe, M.: The triangle closure is a polyhedron. Math. Program. Ser. A 145(1–2), 1–40 (2013). doi: 10.1007/s10107-013-0639-y. Published online 23 February (2013)MATHGoogle Scholar
  5. 5.
    Basu, A., Hildebrand, R., Köppe, M.: Light on the infinite group relaxation I: foundations and taxonomy. 4OR 14(1), 1–40 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Basu, A., Hildebrand, R., Köppe, M.: Light on the infinite group relaxation II: sufficient conditions for extremality, sequences, and algorithms. 4OR 14(2), 107–131 (2016)Google Scholar
  7. 7.
    Basu, A., Hildebrand, R., Köppe, M., Molinaro, M.: A \((k+1)\)-slope theorem for the \(k\)-dimensional infinite group relaxation. SIAM J. Optim. 23(2), 1021–1040 (2013). doi: 10.1137/110848608 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Borozan, V., Cornuéjols, G.: Minimal valid inequalities for integer constraints. Math. Oper. Res. 34, 538–546 (2009). doi: 10.1287/moor.1080.0370.
  9. 9.
    Cornuéjols, G., Margot, F.: On the facets of mixed integer programs with two integer variables and two constraints. Math. Program. 120, 429–456 (2009). doi: 10.1007/s10107-008-0221-1 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cornuéjols, G., Molinaro, M.: A 3-slope theorem for the infinite relaxation in the plane. Math. Program. 142(1–2), 83–105 (2013). doi: 10.1007/s10107-012-0562-7 MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Espinoza, D.G.: Computing with multi-row Gomory cuts. Oper. Res. Lett. 38(2), 115–120 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra, I. Math. Program. 3, 23–85 (1972). doi: 10.1007/BF01585008
  13. 13.
    Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra, II. Math. Program. 3, 359–389 (1972). doi: 10.1007/BF01585008 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gomory, R.E., Johnson, E.L.: T-space and cutting planes. Math. Program. 96, 341–375 (2003). doi: 10.1007/s10107-003-0389-3 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hille, E., Phillips, R.: Functional Analysis and Semi-Groups. American Mathematical Society, Providence (1957)MATHGoogle Scholar
  16. 16.
    Johnson, E.L.: On the group problem for mixed integer programming. Math. Program. Study 2, 137–179 (1974)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Meyer, C.D. (ed.): Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia (2000)Google Scholar
  18. 18.
    Yıldız, S., Cornuéjols, G.: Cut-generating functions for integer variables. Math. Oper. Res. 41, 1381–1403 (2016)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017

Authors and Affiliations

  • Amitabh Basu
    • 1
  • Robert Hildebrand
    • 2
  • Marco Molinaro
    • 3
  1. 1.Department of Applied Mathematics and StatisticsThe Johns Hopkins UniversityBaltimoreUSA
  2. 2.IBM ResearchYorktown HeightsUSA
  3. 3.Computer Science DepartmentPUC-RIORio de JaneiroBrazil

Personalised recommendations