Abstract
Chvátal-Gomory cuts are among the most well-known classes of cutting planes for general integer linear programs (ILPs). In case the constraint multipliers are either 0 or \(\frac {1}{2}\) , such cuts are known as \(\{0,\frac {1}{2}\}\) -cuts. It has been proven by Caprara and Fischetti (Math. Program. 74:221–235, 1996) that separation of \(\{0,\frac {1}{2}\}\) -cuts is \(\mathcal {NP}\) -hard.
In this paper, we study ways to separate \(\{0,\frac {1}{2}\}\) -cuts effectively in practice. We propose a range of preprocessing rules to reduce the size of the separation problem. The core of the preprocessing builds a Gaussian elimination-like procedure. To separate the most violated \(\{0,\frac {1}{2}\}\) -cut, we formulate the (reduced) problem as integer linear program. Some simple heuristic separation routines complete the algorithmic framework.
Computational experiments on benchmark instances show that the combination of preprocessing with exact and/or heuristic separation is a very vital idea to generate strong generic cutting planes for integer linear programs and to reduce the overall computation times of state-of-the-art ILP-solvers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Achterberg, T.: SCIP—a framework to integrate constraint and mixed integer programming. ZIB-Report 04–19, Zuse Institute Berlin (2004). http://www.zib.de/Publications/abstracts/ZR-04-19/
Achterberg, T., Berthold, T., Koch, T., Martin, A., Wolter, K.: SCIP (Solving Constraint Integer Programs) (2006). http://scip.zib.de/
Achterberg, T., Koch, T., Martin, A.: MIPLIB (2003). http://miplib.zib.de
Achterberg, T., Wunderling, R.: Private communication, November 2007
Andreello, G., Caprara, A., Fischetti, M.: Embedding \(\{0,\frac{1}{2}\}\) -cuts in a branch-and-cut framework: A computational study. INFORMS J. Comput. 19(2), 229–238 (2007)
Bixby, R., Reinelt, G.: TSPLIB. http://elib.zib.de/pub/mp-testdata/tsp/tsplib/tsplib.html
Bixby, R.E., Ceria, S., McZeal, C.M., Savelsbergh, M.W.P.: MIPLIB 3.0. http://www.caam.rice.edu/~bixby/miplib/miplib.html
Bonami, P., Cornuéjols, G., Dash, S., Fischetti, M., Lodi, A.: Projected Chvátal–Gomory cuts for mixed integer linear programs. Math. Program. A 113, 241–257 (2008)
Caprara, A., Fischetti, M.: {0,1/2}-Chvátal-Gomory cuts. Math. Program. 74, 221–235 (1996)
Caprara, A., Fischetti, M.: Odd cut-sets, odd cycles, and 0–1/2 Chvátal-Gomory cuts. Ric. Oper. 26, 51–80 (1996)
Caprara, A., Fischetti, M., Letchford, A.N.: On the separation of maximally violated mod-k cuts. Math. Program. 87, 37–56 (2000)
Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Math. 4, 305–337 (1973)
Edmonds, J., Johnson, E.L.: Matching: a well-solved class of integer linear programs. In: Guy, R.K., Hanani, H. (eds.) Combinatorial Structures and Their Applications, pp. 80–92. Gordon and Breach, New York (1970)
Fiorini, S.: \(\{0,\frac{1}{2}\}\) -cuts and the linear ordering problem: Surfaces that define facets. SIAM J. Discrete Math. 20(4), 893–912 (2006)
Fischetti, M., Lodi, A.: Optimizing over the first Chvátal closure. Math. Program. 110(1), 3–20 (2007)
Gentile, C., Ventura, P., Weismantel, R.: Mod-2 cuts generation yields the convex hull of bounded integer feasible sets. SIAM J. Discrete Math. 20(4), 913–919 (2006)
Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bull. Am. Math. Soc. 64, 275–278 (1958)
Gomory, R.E.: An algorithm for integer solutions to linear programs. In: Graves, R.L., Wolfe, P. (eds.) Recent Advances in Mathematical Programming, pp. 269–302. McGraw-Hill, New York (1963)
ILOG. CPLEX version 10.0 (2006). http://www.ilog.com/products/cplex
Koster, A.M.C.A., Zymolka, A.: Stable multi-sets. Math. Methods Oper. Res. 56(1), 45–65 (2002)
Koster, A.M.C.A., Zymolka, A.: On cycles and the stable multi-set polytope. Discrete Optim. 2(3), 241–255 (2005)
Padberg, M.: On the facial structure of set packing polyhedra. Math. Program. 5, 199–215 (1973)
Schrijver, A.: On cutting planes. Ann. Discrete Math. 9, 291–296 (1980)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Koster, A.M.C.A., Zymolka, A. & Kutschka, M. Algorithms to Separate \(\{0,\frac{1}{2}\}\) -Chvátal-Gomory Cuts. Algorithmica 55, 375–391 (2009). https://doi.org/10.1007/s00453-008-9218-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-008-9218-7