Abstract
We present a new method to compute upper bounds of the number of solutions of binary integer programming (BIP) problems. Given a BIP, we create a dynamic programming (DP) table for a redundant knapsack constraint which is obtained by surrogate relaxation. We then consider a Lagrangian relaxation of the original problem to obtain an initial weight bound on the knapsack. This bound is then refined through subgradient optimization. The latter provides a variety of Lagrange multipliers which allow us to filter infeasible edges in the DP table. The number of paths in the final table then provides an upper bound on the number of solutions. Numerical results show the effectiveness of our counting framework on automatic recording and market split problems.
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 - A Framework to Integrate Constraint and Mixed Integer Programming, http://www.zib.de/Publications/abstracts/ZR-04-19/
Ahuja, R.K., Magnati, T.L., Orlin, J.B.: Network Flows. Prentice Hall, Englewood Cliffs (1993)
Birnbaum, E., Lozinskii, E.L.: The Good Old Davis-Putnam Procedure Helps Counting Models. Journal of Artificial Intelligence Research 10, 457–477 (1999)
Cornuejols, G., Dawande, M.: A class of hard small 0-1 programs. In: Bixby, R.E., Boyd, E.A., RÃos-Mercado, R.Z. (eds.) IPCO 1998. LNCS, vol. 1412, pp. 284–293. Springer, Heidelberg (1998)
Danna, E., Fenelon, M., Gu, Z., Wunderling, R.: Generating Multiple Solutions for Mixed Integer Programming Problems. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 280–294. Springer, Heidelberg (2007)
Frangioni, A.: Object Bundle Optimization Package, www.di.unipi.it/optimize/Software/Bundle.html
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1991)
Gomes, C.P., Hoeve, W., Sabharwal, A., Selman, B.: Counting CSP Solutions Using Generalized XOR Constraints. In: 22nd Conference on Artificial Intelligence (AAAI), pp. 204–209 (2007)
Hadzic, T., O’Mahony, E., O’Sullivan, B., Sellmann, M.: Enhanced Inference for the Market Split Problem. In: 21st IEEE International Conference on Tools with Artificial Intelligence (ICTAI), pp. 716–723 (2009)
IBM. IBM CPLEX Reference manual and user manual. V12.1, IBM (2009)
Kroc, L., Sabharwal, A., Selman, B.: Leveraging Belief Propagation, Backtrack Search, and Statistics for Model Counting. In: Perron, L., Trick, M.A. (eds.) CPAIOR 2008. LNCS, vol. 5015, pp. 278–282. Springer, Heidelberg (2008)
Sellmann, M.: Approximated Consistency for Knapsack Constraints. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 679–693. Springer, Heidelberg (2003)
Sellmann, M.: Cost-Based Filtering for Shorter Path Constraints. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 694–708. Springer, Heidelberg (2003)
Sellmann, M.: Theoretical Foundations of CP-based Lagrangian Relaxation. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 634–647. Springer, Heidelberg (2004)
Sellmann, M., Fahle, T.: Constraint Programming Based Lagrangian Relaxation for the Automatic Recording Problem. Annals of Operations Research (AOR), 17–33 (2003)
Sellmann, M.: Approximated Consistency for the Automatic Recording Constraint. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 822–826. Springer, Heidelberg (2005)
Sellmann, M.: Approximated Consistency for the Automatic Recording Constraint. Computers and Operations Research 36(8), 2341–2347 (2009)
Sellmann, M.: ARP: A Benchmark Set for the Automatic Recording Problem maintained, http://www.cs.brown.edu/people/sello/arp-benchmark.html
TIVOtm System, http://www.tivo.com
Trick, M.: A Dynamic Programming Approach for Consistency and Propagation for Knapsack Constraints. In: 3rd Int. Workshop on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems (CPAIOR), pp. 113–124 (2001)
Trick, M.: A Dynamic Programming Approach for Consistency and Propagation for Knapsack Constraints. Annals of Operations Research 118, 73–84 (2003)
Williams, H.P.: Model Building in Mathematical Programming. Wiley, Chicester (1978)
Zanarini, A., Pesant, G.: Solution counting algorithms for constraint-centered search heuristics. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 743–757. Springer, Heidelberg (2007)
Zanarini, A., Pesant, G.: Solution counting algorithms for constraint-centered search heuristics. Constraints 14(3), 392–413
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jain, S., Kadioglu, S., Sellmann, M. (2010). Upper Bounds on the Number of Solutions of Binary Integer Programs. In: Lodi, A., Milano, M., Toth, P. (eds) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. CPAIOR 2010. Lecture Notes in Computer Science, vol 6140. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13520-0_24
Download citation
DOI: https://doi.org/10.1007/978-3-642-13520-0_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13519-4
Online ISBN: 978-3-642-13520-0
eBook Packages: Computer ScienceComputer Science (R0)