Advertisement

Mathematical Programming

, Volume 124, Issue 1–2, pp 45–68 | Cite as

Small Chvátal Rank

  • Tristram BogartEmail author
  • Annie Raymond
  • Rekha Thomas
Full Length Paper Series B

Abstract

We propose a variant of the Chvátal-Gomory procedure that will produce a sufficient set of facet normals for the integer hulls of all polyhedra {x : A x ≤ b} as b varies. The number of steps needed is called the small Chvátal rank (SCR) of A. We characterize matrices for which SCR is zero via the notion of supernormality which generalizes unimodularity. SCR is studied in the context of the stable set problem in a graph, and we show that many of the well-known facet normals of the stable set polytope appear in at most two rounds of our procedure. Our results reveal a uniform hypercyclic structure behind the normals of many complicated facet inequalities in the literature for the stable set polytope. Lower bounds for SCR are derived both in general and for polytopes in the unit cube.

Keywords

Integer programming Cutting planes Chvátal rank Supernormality Stable set polytopes 

Mathematics Subject Classification (2000)

90C10 90C27 90C57 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon N., Vu V.: Anti-Hadamard matrices, coin weighing, threshold gates and indecomposable hypergraphs. J. Comb. Theory Ser. A 79(1), 133–160 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bach E., Shallit J.: Algorithmic Number Theory, Vol. 1: Efficient Algorithms, Foundations of Computing Series. MIT Press, Cambridge, MA (1996)Google Scholar
  3. 3.
    Barvinok A., Woods K.: Short rational generating functions for lattice point problems. J. Am. Math. Soc. 16, 957–979 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Balas E., Ceria S., Cornuéjols G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58, 295–324 (1993)CrossRefGoogle Scholar
  5. 5.
    Bruns, W., Ichim, B.: NORMALIZ. Computing normalizations of affine semigroups. With contributions by C. Söger. Available at http://www.math.uos.de/normaliz
  6. 6.
    Chvátal V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discret. Math. 4, 305–337 (1973)zbMATHCrossRefGoogle Scholar
  7. 7.
    Eisenbrand F., Schulz A.S.: Bounds on the Chvátal rank of polytopes in the 0/1 cube. Combinatorica 23(2), 245–261 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Galluccio A., Sassano A.: The rank facets of the stable set polytope for claw-free graphs. J. Comb. Theory Ser. B 69(1), 1–38 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Giles R., Trotter L.E. Jr.: On stable set polyhedra for K 1,3-free graphs. J. Comb. Theory Ser. B 31(3), 313–326 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Grötschel M., Lovász L., Schrijver A.: Geometric Algorithms and Combinatorial Optimization. Volume 2 of Algorithms and Combinatorics. 2nd edn. Springer, Berlin (1993)Google Scholar
  11. 11.
    Hoşten S., Maclagan D., Sturmfels B.: Supernormal vector configurations. J. Algebraic Comb. 19(3), 297–313 (2004)zbMATHCrossRefGoogle Scholar
  12. 12.
    Liebling T.M., Oriolo G., Spille B., Stauffer G.: On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs. Math. Methods Oper. Res. 59(1), 25–35 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lipták L., Lovász L.: Facets with fixed defect of the stable set polytope. Math. Program. 88(1, Ser. A), 33–44 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lovász L., Schrijver A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1(2), 166–190 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Maclagan D., Thomas R.R.: The toric Hilbert scheme of a rank two lattice is smooth and irreducible. J. Comb. Theory Ser. A 104, 29–48 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Schrijver A.: Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York (1986)Google Scholar
  17. 17.
    Schrijver, A.: Combinatorial optimization. Polyhedra and efficiency. Vol. A, volume 24 of Algorithms and Combinatorics. Springer, Berlin (2003) (Paths, flows, matchings, Chapters 1–38)Google Scholar
  18. 18.
    Schrijver, A.: Combinatorial optimization. Polyhedra and efficiency. Vol. A, volume 24 of Algorithms and Combinatorics. Springer, Berlin (2003) (Matroids, trees, stable sets, Chapters 39–69)Google Scholar
  19. 19.
    Seymour P.D.: Decomposition of regular matroids. J. Comb. Theory Ser. B 28(3), 305–359 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Sherali H.D., Adams W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discret. Math. 3(3), 411–430 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Stephen T., Tunçel L.: On a representation of the matching polytope via semidefinite liftings. Math. Oper. Res. 24(1), 1–7 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Ziegler G.M.: Lectures on 0/1-polytopes. Polytopes—combinatorics and computation, Oberwolfach, 1997, volume 29 of DMV Sem, pp. 1–41. Birkhäuser, Basel (2000)Google Scholar

Copyright information

© Springer and Mathematical Programming Society 2010

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada
  2. 2.Berlin Mathematical SchoolTechnical UniversityBerlinGermany
  3. 3.Department of MathematicsUniversity of WashingtonSeattleUSA

Personalised recommendations