Abstract
The paper investigates the relationship between counting the lattice points belonging to an hyperplane and the separation of Chvátal-Gomory cutting planes. In particular, we show that counting can be exploited in two ways: (i) to strengthen the cuts separated, e.g., by Gomory classical procedure, and (ii) to heuristically evaluate the effectiveness of those cuts and possibly select only a subset of them. Empirical results on a small set of 0-1 Integer Programming instances are presented.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Achterberg, T.: SCIP: Solving Constraint Integer Programs. Mathematical Programming Computation 1, 1–32 (2008)
Achterberg, T.: LP Basis Selection and Cutting Planes. Research Talk @ MIP (2010), http://www2.isye.gatech.edu/mip2010/program/program.pdf
Andreello, G., Caprara, A., Fischetti, M.: Embedding \(\{0,\frac{1}{2}\}\)-Cuts in a Branch-and-Cut Framework: A Computational Study. INFORMS Journal on Computing 19, 229–238 (2007)
Balas, E., Ceria, S., Cornuéjols, G., Natraj, N.: Gomory Cuts Revisited. Operations Research Letters 19, 1–9 (1996)
Caprara, A., Lodi, A., Scheinberg, K. (eds.): Counting and Estimating Lattice Points. Optima, vol. (81), http://www.mathprog.org/Optima-Issues/optima81.pdf
Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Mathematics 4, 305–337 (1973)
Dunkel, J.: On Gomory-Chvátal Cutting Planes, the Elementary Closure, and a Strengthened Closure for Polytopes in the Unit Cube. PhD thesis. MIT, Cambridge, MA (2011)
Eisenbrand, F.: On the membership problem for the elementary closure of a polyhedron. Combinatorica 19, 297–300 (1999)
Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bulletin of the AMS 64, 275–278 (1958)
Linderoth, J.T., Lodi, A.: MILP Software. In: Cochran, J.J. (ed.) Wiley Encyclopedia of Operations Research and Management Science, vol. 5, pp. 3239–3248. Wiley, Chichester (2011)
Lodi, A.: MIP computation. In: Jünger, M., Liebling, T.M., Naddef, D., Nemhauser, G.L., Pulleyblank, W.R., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds.) 50 Years of Integer Programming 1958-2008, pp. 619–645. Springer, Heidelberg (2009)
Pesant, G., Quimper, C.-G.: Counting solutions of knapsack constraints. In: Perron, L., Trick, M.A. (eds.) CPAIOR 2008. LNCS, vol. 5015, pp. 203–217. Springer, Heidelberg (2008)
Trick, M.A.: A Dynamic Programming Approach for Consistency and Propagation for Knapsack Constraints. Annals of Operations Research 118, 73–84 (2003)
Pryor, J., Chinneck, J.W.: Faster Integer-Feasibility in Mixed-Integer Linear Programs by Branching to Force Change. Computers & OR 38, 1143–1152 (2011)
Wesselmann, F., Suhl, U.H.: Implementation techniques for cutting plane management and selection. Technical Report Universität Paderborn (2007)
Zanarini, A., Pesant, G.: Solution counting algorithms for constraint-centered search heuristics. Constraints 14, 392–413 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lodi, A., Pesant, G., Rousseau, LM. (2011). On Counting Lattice Points and Chvátal-Gomory Cutting Planes. In: Achterberg, T., Beck, J.C. (eds) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. CPAIOR 2011. Lecture Notes in Computer Science, vol 6697. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21311-3_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-21311-3_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21310-6
Online ISBN: 978-3-642-21311-3
eBook Packages: Computer ScienceComputer Science (R0)