Abstract
The Time-Invariant Incremental Knapsack problem (IIK) is a generalization of Maximum Knapsack to a discrete multi-period setting. At each time, capacity increases and items can be added, but not removed from the knapsack. The goal is to maximize the sum of profits over all times. IIK models various applications including specific financial markets and governmental decision processes. IIK is strongly NP-Hard [2] and there has been work [2, 3, 6, 13, 15] on giving approximation algorithms for some special cases. In this paper, we settle the complexity of IIK by designing a PTAS based on rounding a disjunctive formulation, and provide several extensions of the technique.
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Notes
- 1.
See the full version of the paper [4] for a discussion on disjunctive programming.
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Acknowledgements
We thank Andrey Kupavskii for valuable combinatorial insights on the topic. Yuri Faenza’s research was partially supported by the SNSF Ambizione fellowship PZ00P2\(\_\)154779 Tight formulations of 0 / 1 problems. Some of the work was done when Igor Malinović visited Columbia University partially funded by a gift from the SNSF.
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Faenza, Y., Malinovic, I. (2018). A PTAS for the Time-Invariant Incremental Knapsack Problem. In: Lee, J., Rinaldi, G., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2018. Lecture Notes in Computer Science(), vol 10856. Springer, Cham. https://doi.org/10.1007/978-3-319-96151-4_14
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