A separation algorithm for knapsack polytope is proposed. This algorithm has been used in the branch-and-cut method for solving the generalized assignment problem and the capacitated p-median problem. The computational experiment on the test instances has shown that this method is highly competitive in comparison with the existing approaches.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
S. Martello and P. Toth, Knapsack Problems — Algorithms and Computer Implementations (Wiley, New York, 1990).
H. Kellerer, U. Pferschy, and D. Pisinger, Knapsack Problems (Springer, Berlin, 2005).
G. L. Nemhauser and L. A. Wolsey, Integer and Combinatorial Optimization (Wiley, New York, 1988).
E. Balas, “Facets of the Knapsack Polytope,” Mathmatical Programming 8, 146–164 (1975).
ILOG CPLEX 10.0 Reference Manual. Sunnyvale: ILOG Inc, 2006.
Xpress-MP Optimizer 18.0 Reference Manual. Leamington Spa: Dashoptimzation, 2007.
P. Avella, M. Boccia, and I. Vasilyev, “A Computational Study of Exact Knapsack Separation for the Generalized Assignment Problem,” Computational Optimization and Applications (2008).
M. Boccia, A. Sforza, C. Sterle, et al., “A Cut and Branch Approach for the Capacitated P-Median Problem Based on Fenchel Cutting Planes,” J. Mathematical Modelling and Algorithms 7, 43–58 (2007).
A. Ceselli and G. Righini, “A Branch and Price Algorithm for the Capacitated P-Median Problem,” Networks 45(3), 125–142 (2005).
A. Pigatti, Poggi de Aragao M., Uchoa E., “Stabilized Branch-and-Cut-and-Price for the Generalized Assignment Problem,” in Proceedings of 2nd Brasilian Symposium on Graphs, Algorithms and Combinatorics, Electronic Notes in Discrete Mathematics, Rio de Janeiro, Brasilia, 2005, Vol. 19, pp. 389–395.
A. Ceselli, “Two exact algorithms for the capacitated pmedian problem,” 4OR., 1, 319–340 (2003).
E. A. Boyd, “Generating Fenchel Cutting Planes for Knapsack Polyhedra,” SIAM J. Optimization 3, 734–750 (1993).
E. A. Boyd, “Fenchel Cutting Planes for Integer Programming,” Oper. Res. 42, 53–62 (1994).
E. A. Boyd, “On the Convergence Fenchel Cutting Planes in Mixed-Integer Programming,” SIAM J. Optimization 5, 421–435 (1995).
R. Fukasawa and M. Goycoolea, “On the Exact Separation of Mixed Integer Knapsack Cuts,” in Proceedings of 12th Integer Programming and Combinatorial Optimization Conference (Lecture Notes in Computer Science, New York, 2007), Vol. 4513, pp. 225–239.
K. Kaparis and A. N. Letchford, Separation Algorithms for, 0–1 Knapsack Polytopes, Working paper, http://www.lancs.ac.uk/staff/letchfoa/publications.htm.2007.
M. W. Padberg, “On the Facial Structure of Set Packing Polyhedra,” Mathematical Programming 5, 199–215 (1973).
L. A. Wolsey, “Facets and Strong Valid Inequalities for Integer Programs,” Oper. Res. 24, 367–372 (1976).
D. Pisinger, “A Minimal Algorithm for the 0–1 Knapsack Problem,” Oper. Res. 46, 758–767 (1995).
V. Maniezzo, A. Mingozzi, and R. Baldacci, “A Biotomic Approach to the Capacitated P-Median Problem,” J. Heuristics 4, 263–280 (1998).
R. Baldacci, E. Hadjiconstantinou, V. Maniezzo, et al., “A New Method for Solving Capacitated Location Problems Based on a Set Partitioning Approach,” Comput. Oper. Res. 29, 365–386 (2002).
J. E. Beasley, “Or-Library: Distributing Test Problems by Electronic Mail,” J. Operational Research Society 41(11), 1069–1072 (1990).
Original Russian Text © I.L. Vasil’ev, 2009, published in Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, 2009, No. 1, pp. 74–81.
About this article
Cite this article
Vasil’ev, I.L. A cutting plane method for knapsack polytope. J. Comput. Syst. Sci. Int. 48, 70–77 (2009). https://doi.org/10.1134/S1064230709010067
- Knapsack Problem
- System Science International
- Valid Inequality
- Separation Problem
- Separation Algorithm