Abstract
We study the d-dimensional knapsack problem in the data streaming model. The knapsack is modelled as a d-dimensional integer vector of capacities. For simplicity, we assume that the input is scaled such that all capacities are 1. There is an input stream of n items, each item is modelled as a d-dimensional integer column of non-negative integer weights and a scalar profit. The input instance has to be processed in an online fashion using sub-linear space. After the items have arrived, an approximation for the cost of an optimal solution as well as a template for an approximate solution is output.
Our algorithm achieves an approximation ratio \((2(\frac{1}{2}+\sqrt{2 d+\frac{1}{4}}))^{-1}\) using space O(2O(d) ·logd + 1 d ·logd + 1 Δ·logn) bits, where \(\{\frac{1}{\Delta}, \frac{2}{\Delta}, \dots, 1\}\), Δ ≥ 2 is the set of possible profits and weights in any dimension. We also show that any data streaming algorithm for the t(t − 1)-dimensional knapsack problem that uses space \(o(\sqrt{\Delta}/t^2)\) cannot achieve an approximation ratio that is better than 1/t. Thus, even using space Δγ, for γ< 1/2, i.e. space polynomial in Δ, will not help to break the \(1/t \approx 1/\sqrt{d}\) barrier in the approximation ratio.
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Chakrabarti, A., Khot, S., Sun, X.: Near-Optimal Lower Bounds on the Multi-Party Communication Complexity of Set Disjointness. In: Proceedings of International Conference on Computational Complexity, pp. 107–117 (2003)
Chekuri, C., Khanna, S.: On multi-dimensional packing problems. In: Proceedings of the 10th ACM-SIAM Symposium on Discrete Algorithms, pp. 185–194 (1999)
Diestel, R.: Graphentheorie. Springer, Heidelberg (2006)
Ibarra, O.H., Kim, C.E.: Fast Approximation Algorithms for the Knapsack and the Sum of Subset Problems. J. ACM 22(4) (October 1975)
Iwama, K., Taketomi, S.: Removable online knapsack problems. In: Proc. of the 29th Intl. Conf. on Automata, Languages and Programming, pp. 293–305 (2002)
Iwama, K., Zhang, G.: Optimal Resource Augmentations for Online Knapsack. In: Proceedings of the 10th Intl. Workshop on Approximation Algorithms for Combinatorial Optimization Problems, pp. 180–188 (2007)
Karp, R.: Reducibility among Combinatorial Problems. In: Complexity of Computer Computations, pp. 85–103 (1972)
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© 2009 Springer-Verlag Berlin Heidelberg
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Ganguly, S., Sohler, C. (2009). d-Dimensional Knapsack in the Streaming Model. In: Fiat, A., Sanders, P. (eds) Algorithms - ESA 2009. ESA 2009. Lecture Notes in Computer Science, vol 5757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04128-0_42
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DOI: https://doi.org/10.1007/978-3-642-04128-0_42
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04127-3
Online ISBN: 978-3-642-04128-0
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