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Reverse Split Rank

  • Michele Conforti
  • Alberto Del Pia
  • Marco Di Summa
  • Yuri Faenza
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8494)

Abstract

The reverse split rank of an integral polytope P is defined as the supremum of the split ranks of all rational polyhedra whose integer hull is P. Already in ℝ3 there exist polytopes with infinite reverse split rank. We give a geometric characterization of the integral polytopes in ℝ n with infinite reverse split rank.

Keywords

Convex Hull Linear Subspace Integer Point Discrete Apply Mathematic Rational Polyhedron 
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.

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References

  1. 1.
    Averkov, G., Conforti, M., Del Pia, A., Di Summa, M., Faenza, Y.: On the convergence of the affine hull of the Chvátal-Gomory closures. SIAM Journal on Discrete Mathematics 27, 1492–1502 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Balas, E.: Disjunctive programming: Properties of the convex hull of feasible points. Discrete Applied Mathematics 89, 3–44 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Barvinok, A.: A Course in Convexity, Grad. Stud. Math. 54, AMS, Providence, RI (2002)Google Scholar
  4. 4.
    Basu, A., Conforti, M., Cornuéjols, G., Zambelli, G.: Maximal lattice-free convex sets in linear subspaces. Mathematics of Operations Research 35, 704–720 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Basu, A., Cornuéjols, G., Margot, F.: Intersection cuts with infinite split rank. Mathematics of Operations Research 37, 21–40 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Conforti, M., Del Pia, A., Di Summa, M., Faenza, Y., Grappe, R.: Reverse Chvátal–Gomory rank. In: Goemans, M., Correa, J. (eds.) IPCO 2013. LNCS, vol. 7801, pp. 133–144. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  7. 7.
    Conforti, M., Del Pia, A., Di Summa, M., Faenza, Y., Grappe, R.: Reverse Chvátal–Gomory rank (submitted, 2014)Google Scholar
  8. 8.
    Cook, W., Coullard, C.R., Túran, G.: On the complexity of cutting-plane proofs. Discrete Applied Mathematics 18, 25–38 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Cook, W., Kannan, R., Schrijver, A.: Chvátal closures for mixed integer programming problems. Mathematical Programming 174, 155–174 (1990)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Del Pia, A.: On the rank of disjunctive cuts. Mathematics of Operations Research 37, 372–378 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Dey, S.S., Louveaux, Q.: Split rank of triangle and quadrilateral inequalities. Mathematics of Operations Research 36, 432–461 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Eisenbrand, F., Schulz, A.S.: Bounds on the Chvátal rank of polytopes in the 0/1 cube. Combinatorica 23, 245–261 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Kannan, R., Lovasz, L.: Covering minima and lattice-point-free convex bodies. Ann. Math., Second Series 128, 577–602 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Khintchine, A.: A quantitative formulation of Kronecker’s theory of approximation. Izv. Acad. Nauk SSSR, Ser. Mat. 12, 113–122 (1948) (in Russian)Google Scholar
  15. 15.
    Lovász, L.: Geometry of Numbers and Integer Programming. In: Iri, M., Tanabe, K. (eds.) Mathematical Programming: Recent Developements and Applications, pp. 177–210. Kluwer (1989)Google Scholar
  16. 16.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley-Interscience, New York (1988)zbMATHGoogle Scholar
  17. 17.
    Pokutta, S., Stauffer, G.: Lower bounds for the Chvátal–Gomory rank in the 0/1 cube. Operations Research Letters 39, 200–203 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  19. 19.
    Rothvoß, T., Sanitá, L.: 0/1 polytopes with quadratic chvátal rank. In: Goemans, M., Correa, J. (eds.) IPCO 2013. LNCS, vol. 7801, pp. 349–361. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  20. 20.
    Schrijver, A.: On cutting planes. Annals of Discrete Mathematics 9, 291–296 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Schrijver, A.: Theory of Linear and Integer Programming. John Wiley (1986)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Michele Conforti
    • 1
  • Alberto Del Pia
    • 2
  • Marco Di Summa
    • 1
  • Yuri Faenza
    • 3
  1. 1.Dipartimento di MatematicaUniversità degli Studi di PadovaItaly
  2. 2.Business Analytics and Mathematical Sciences DepartmentIBM Watson Research CenterYorktown HeightsUSA
  3. 3.DISOPTInstitut de mathématiques d’analyse et applications, EPFLSwitzerland

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