Abstract
In this paper we propose an improved efficient approximation scheme for the multiple knapsack problem (MKP). Given a set \({\mathcal A}\) of n items and set \({\mathcal B}\) of m bins with possibly different capacities, the goal is to find a subset \(S \subseteq{\mathcal A}\) of maximum total profit that can be packed into \({\mathcal B}\) without exceeding the capacities of the bins. Chekuri and Khanna presented a PTAS for MKP with arbitrary capacities with running time \(n^{O(1/\epsilon^8 \log(1/\epsilon))}\). Recently we found an efficient polynomial time approximation scheme (EPTAS) for MKP with running time \(2^{O(1/\epsilon^5 \log(1/\epsilon))} poly(n)\). Here we present an improved EPTAS with running time \(2^{O(1/\epsilon \log^4(1/\epsilon))} + poly(n)\). If the integrality gap between the ILP and LP objective values for bin packing with different sizes is bounded by a constant, the running time can be further improved to \(2^{O(1/\epsilon \log^2(1/\epsilon))} + poly(n)\).
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References
Caprara, A., Kellerer, H., Pferschy, U.: The multiple subset sum problem. SIAM Journal of Optimization 11, 308–319 (2000)
Chekuri, C., Khanna, S.: A PTAS for the multiple knapsack problem. SIAM Journal on Computing 35, 713–728 (2006)
Fellows, M.R.: Blow-Ups, Win/Win’s, and Crown Rules: Some New Directions in FPT. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 1–12. Springer, Heidelberg (2003)
Jansen, K.: Parameterized approximation scheme for the multiple knapsack problem. SIAM Journal on Computing 39, 1392–1412 (2009)
Karmarkar, N., Karp, R.M.: An efficient approximation scheme for the one-dimensional bin-packing problem. In: Proceedings of the 23rd IEEE Symposium on Foundations of Computer Science, FOCS 1982, pp. 312–320 (1982)
Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Berlin (2004)
Kenyon, C., Remila, E.: Approximate strip packing. Mathematics of Operations Research 25, 645–656 (2000)
Marx, D.: Parametrized complexity and approximation algorithms. The Computer Journal 51, 60–78 (2008)
Scheithauer, G., Terno, J.: Theoretical investigations on the modified integer round-up property for the one-dimensional cutting stock problem. European Journal of Operational Research 20, 93–100 (1997)
Shmonin, G.: Parameterised integer programming, integer cones, and related problems, PhD thesis, Universität Paderborn (2007)
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Jansen, K. (2012). A Fast Approximation Scheme for the Multiple Knapsack Problem. In: Bieliková, M., Friedrich, G., Gottlob, G., Katzenbeisser, S., Turán, G. (eds) SOFSEM 2012: Theory and Practice of Computer Science. SOFSEM 2012. Lecture Notes in Computer Science, vol 7147. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27660-6_26
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DOI: https://doi.org/10.1007/978-3-642-27660-6_26
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