Approximation of Corner Polyhedra with Families of Intersection Cuts

  • Gennadiy Averkov
  • Amitabh Basu
  • Joseph PaatEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10328)


We study the problem of approximating the corner polyhedron using intersection cuts derived from families of lattice-free sets. In particular, we look at the problem of characterizing families that approximate the corner polyhedron up to a constant factor in fixed dimension n (the constant depends on n). The literature already contains several results in this direction. In this paper, we use the maximum number of facets of a lattice-free set in a family as a measure of its complexity and precisely characterize the level of complexity of a family required for constant factor approximations. As one of the main results, we show that for each natural number n, a corner polyhedron for n integer variables is approximated by intersection cuts from lattice-free sets with at most i facets up to a constant factor (depending only on n) if \(i> 2^{n-1}\) and that no such approximation is possible if \(i \le 2^{n-1}\). When the approximation factor is allowed to depend on the denominator of the underlying fractional point of the corner polyhedron, we show that the threshold is \(i > n\) versus \(i \le n\). The tools introduced for proving such results are of independent interest for studying intersection cuts.


Relative Strength Gauge Function Constant Factor Approximation Recession Cone Standard Basis Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Andersen, K., Louveaux, Q., Weismantel, R., Wolsey, L.A.: Inequalities from two rows of a simplex tableau. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 1–15. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-72792-7_1 CrossRefGoogle Scholar
  2. 2.
    Andersen, K., Wagner, C., Weismantel, R.: On an analysis of the strength of mixed-integer cutting planes from multiple simplex tableau rows. SIAM J. Optim. 20(2), 967–982 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Averkov, G., Krümpelmann, J., Weltge, S.: Notions of maximality for integral lattice-free polyhedra: the case of dimension three (2015).
  4. 4.
    Averkov, G., Wagner, C., Weismantel, R.: Maximal lattice-free polyhedra: finiteness and an explicit description in dimension three. Math. Oper. Res. 36(4), 721–742 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Awate, Y., Cornuéjols, G., Guenin, B., Tuncel, L.: On the relative strength of families of intersection cuts arising from pairs of tableau constraints in mixed integer programs. Math. Program. 150, 459–489 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barvinok, A.: A Course in Convexity. American Mathematical Society (2002)Google Scholar
  7. 7.
    Basu, A., Bonami, P., Cornuéjols, G., Margot, F.: On the relative strength of split, triangle and quadrilateral cuts. Math. Program. Ser. A 126, 281–314 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Basu, A., Conforti, M., Cornuéjols, G., Zambelli, G.: Maximal lattice-free convex sets in linear subspaces. Math. Oper. Res. 35, 704–720 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Borozan, V., Cornuéjols, G.: Minimal valid inequalities for integer constraints. Math. Oper. Res. 34, 538–546 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Conforti, M., Cornuéjols, G., Zambelli, G.: Integer programming, vol. 271. Springer, Switzerland (2014)zbMATHGoogle Scholar
  11. 11.
    Dey, S.S., Wolsey, L.A.: Lifting integer variables in minimal inequalities corresponding to lattice-free triangles. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 463–475. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-68891-4_32 CrossRefGoogle Scholar
  12. 12.
    Gomory, R.E.: Some polyhedra related to combinatorial problems. Linear Algebra Appl. 2(4), 451–558 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lovász, L.: Geometry of numbers and integer programming. In: Iri, M., Tanabe, K. (eds.) Mathematical Programming: State of the Art, pp. 177–201. Mathematical Programming Society (1989)Google Scholar
  14. 14.
    Meyer, R.: On the existence of optimal solutions to integer and mixed-integer progamming problems. Math. Program. 7, 223–235 (1974)CrossRefzbMATHGoogle Scholar
  15. 15.
    Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory, vol. 44. Cambridge University Press, Cambridge (2014)zbMATHGoogle Scholar
  16. 16.
    Zambelli, G.: On degenerate multi-row gomory cuts. Oper. Res. Lett. 37(1), 21–22 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics, Institute of Mathematical OptimizationUniversity of MagdeburgMagdeburgGermany
  2. 2.Department of Applied Mathematics and StatisticsJohns Hopkins UniversityBaltimoreUSA

Personalised recommendations