Abstract
The Multiple Subset Sum Problem (MSSP) is the variant of bin packing in which the number of bins is given and one would like to maximize the overall weight of the items packed in the bins. The problem is also a special case of the multiple knapsack problem in which all knapsacks have the same capacity and the item profits and weights coincide. Recently, polynomial time approximation schemes have been proposed for MSSP and its generalizations, see A. Caprara, H. Kellerer, and U. Pferschy (SIAM J. on Optimization, Vol. 11, pp. 308–319, 2000; Information Processing Letters, Vol. 73, pp. 111–118, 2000), C. Chekuri and S. Khanna (Proceedings of SODA 00, 2000, pp. 213–222), and H. Kellerer (Proceedings of APPROX, 1999, pp. 51–62). However, these schemes are only of theoretical interest, since they require either the solution of huge integer linear programs, or the enumeration of a huge number of possible solutions, for any reasonable value of required accuracy. In this paper, we present a polynomial-time 3/4-approximation algorithm which runs fast also in practice. Its running time is linear in the number of items and quadratic in the number of bins. The “core” of the algorithm is a procedure to pack triples of “large” items into the bins. As a byproduct of our analysis, we get the approximation guarantee for a natural greedy heuristic for the 3-Partitioning Problem.
Similar content being viewed by others
References
Caprara, A., H. Kellerer, and U. Pferschy. (2000). “The Multiple Subset Sum Problem.” SIAM Journal on Opti-mization 11, 308–319.
Caprara, A., H. Kellerer, and U. Pferschy. (2000). “A PTAS for the Multiple Subset Sum Problem with Different Knapsack Capacities.” Information Processing Letters 73, 111–118.
Caprara, A., H. Kellerer, and U. Pferschy. (2000). “A 3/4-Approximation Algorithm for Multiple Subset Sum.” Research Report OR/00/7 DEIS. Available at http://www.or.deis.unibo.it/techrep.html.
Chekuri, C. and S. Khanna. (2000). “A PTAS for the Multiple Knapsack Problem.” In Proceedings of SODA 00, pp. 213–222.
Dawande, M., J. Kalagnanam, P. Keskinocak, F.S. Salman, and R. Ravi. (2000). “Approximation Algorithms for the Multiple Knapsack Problem with Assignment Restrictions.” Journal of Combinatorial Optimization 4, 171–186.
Dell'Amico, M. and S. Martello. (1999). “Reduction of the Three-Partition Problem.” Journal of Combinatorial Optimization 3, 17–30.
Garey, M.R. and D.S. Johnson. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. San Francisco: W.H. Freeman.
Kellerer, H. (1999). “A Polynomial Approximation Scheme for the Multiple Knapsack Problem.” In Proceedings of APPROX 99, Lecture Notes in Computer Science, Vol. 1671, pp. 51–62, Berlins: Springer.
Martello, S. and P. Toth. (1990). Knapsack Problems: Algorithms and Computer Implementations. Chichester: J. Wiley & Sons.
Wirsching, G. Personal communication.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Caprara, A., Kellerer, H. & Pferschy, U. A 3/4-Approximation Algorithm for Multiple Subset Sum. Journal of Heuristics 9, 99–111 (2003). https://doi.org/10.1023/A:1022584312032
Issue Date:
DOI: https://doi.org/10.1023/A:1022584312032