A 3/4-Approximation Algorithm for Multiple Subset Sum

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.