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Submodular Maximization with Uncertain Knapsack Capacity

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LATIN 2018: Theoretical Informatics (LATIN 2018)

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Abstract

We consider the maximization problem of monotone submodular functions under an uncertain knapsack constraint. Specifically, the problem is discussed in the situation that the knapsack capacity is not given explicitly and can be accessed only through an oracle that answers whether or not the current solution is feasible when an item is added to the solution. Assuming that cancellation of an item is allowed when it overflows the knapsack capacity, we discuss the robustness ratios of adaptive policies for this problem, which are the worst case ratios of the objective values achieved by the output solutions to the optimal objective values. We present a randomized policy of robustness ratio \((1-1/e)/2\), and a deterministic policy of robustness ratio \(2(1-1/e)/21\). We also consider a universal policy that chooses items following a precomputed sequence. We present a randomized universal policy of robustness ratio \((1-1/\root 4 \of {e})/2\). When the cancellation is not allowed, no randomized adaptive policy achieves a constant robustness ratio. Because of this hardness, we assume that a probability distribution of the knapsack capacity is given, and consider computing a sequence of items that maximizes the expected objective value. We present a polynomial-time randomized algorithm of approximation ratio \((1-1/\root 4 \of {e})/4-\epsilon \) for any small constant  \(\epsilon >0\).

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Acknowledgement

The first author is supported by JSPS KAKENHI Grant Number JP16K16005. The second author is supported by JSPS KAKENHI Grant Number JP17K12646 and JST ERATO Grant Number JPMJER1201, Japan. The third author is supported by JSPS KAKENHI Grant Number JP17K00040 and JST ERATO Grant Number JPMJER1201, Japan.

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Correspondence to Yasushi Kawase .

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Kawase, Y., Sumita, H., Fukunaga, T. (2018). Submodular Maximization with Uncertain Knapsack Capacity. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_48

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  • DOI: https://doi.org/10.1007/978-3-319-77404-6_48

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-77403-9

  • Online ISBN: 978-3-319-77404-6

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