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.
Similar content being viewed by others
References
Alon N., Vu V.: Anti-Hadamard matrices, coin weighing, threshold gates and indecomposable hypergraphs. J. Comb. Theory Ser. A 79(1), 133–160 (1997)
Bach E., Shallit J.: Algorithmic Number Theory, Vol. 1: Efficient Algorithms, Foundations of Computing Series. MIT Press, Cambridge, MA (1996)
Barvinok A., Woods K.: Short rational generating functions for lattice point problems. J. Am. Math. Soc. 16, 957–979 (2003)
Balas E., Ceria S., Cornuéjols G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58, 295–324 (1993)
Bruns, W., Ichim, B.: NORMALIZ. Computing normalizations of affine semigroups. With contributions by C. Söger. Available at http://www.math.uos.de/normaliz
Chvátal V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discret. Math. 4, 305–337 (1973)
Eisenbrand F., Schulz A.S.: Bounds on the Chvátal rank of polytopes in the 0/1 cube. Combinatorica 23(2), 245–261 (2003)
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)
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)
Grötschel M., Lovász L., Schrijver A.: Geometric Algorithms and Combinatorial Optimization. Volume 2 of Algorithms and Combinatorics. 2nd edn. Springer, Berlin (1993)
Hoşten S., Maclagan D., Sturmfels B.: Supernormal vector configurations. J. Algebraic Comb. 19(3), 297–313 (2004)
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)
Lipták L., Lovász L.: Facets with fixed defect of the stable set polytope. Math. Program. 88(1, Ser. A), 33–44 (2000)
Lovász L., Schrijver A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1(2), 166–190 (1991)
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)
Schrijver A.: Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York (1986)
Schrijver, A.: Combinatorial optimization. Polyhedra and efficiency. Vol. A, volume 24 of Algorithms and Combinatorics. Springer, Berlin (2003) (Paths, flows, matchings, Chapters 1–38)
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)
Seymour P.D.: Decomposition of regular matroids. J. Comb. Theory Ser. B 28(3), 305–359 (1980)
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)
Stephen T., Tunçel L.: On a representation of the matching polytope via semidefinite liftings. Math. Oper. Res. 24(1), 1–7 (1999)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
All authors were partially supported by NSF grant DMS-0401047 and the Robert R. and Elaine K. Phelps Endowment at the University of Washington.
Rights and permissions
About this article
Cite this article
Bogart, T., Raymond, A. & Thomas, R. Small Chvátal Rank. Math. Program. 124, 45–68 (2010). https://doi.org/10.1007/s10107-010-0370-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-010-0370-x