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Nonsubmodular Maximization with Knapsack Constraint via Multilinear Extension

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Parallel Architectures, Algorithms and Programming (PAAP 2020)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1362))

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Abstract

For the problem of maximizing a monotone submodular function subject to knapsack constraint, there is a \((1-1/e-\epsilon )\)-approximation algorithm running a nearly-linear time. In this paper, we consider the case that the objective function is nonsubmodular. We propose an approximation algorithm with approximation ratio

$$ \kappa \left( 1+\frac{\epsilon }{\kappa ^{2}(1-\epsilon )} \right) ^{-1} \left( 1- e^{-\varOmega \left( \epsilon ^{2} / \lambda \right) }\right) \left( 1-e^{-\kappa ^{3}}-O(\epsilon ) \right) , $$

and complexity \(\tilde{O} (\frac{1}{1-\tau } n^2 (\log n)^{\frac{1}{\epsilon }+2}),\) where \(\kappa \) is the continuous submodularity ratio, \(\tau \) is the curvature and \(\lambda \) is the largest weight. The technology of our algorithm is using continuous greedy to get a fractional solution and then rounding it with the contention resolution scheme.

Supported by National Natural Science Foundation of China (No. 12001335) and Shandong Provincial Natural Science Foundation, China (Nos. ZR2020MA029, ZR2019PA004).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, People’s Republic of China

    Jiachen Ju, Min Li, Jianxin Liu, Qian Liu & Yang Zhou

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  1. Jiachen Ju
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  2. Min Li
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  3. Jianxin Liu
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  4. Qian Liu
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  5. Yang Zhou
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Corresponding author

Correspondence to Qian Liu .

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Editors and Affiliations

  1. Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China

    Li Ning

  2. Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China

    Vincent Chau

  3. The University of Hong Kong, Hong Kong, China

    Francis Lau

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Ju, J., Li, M., Liu, J., Liu, Q., Zhou, Y. (2021). Nonsubmodular Maximization with Knapsack Constraint via Multilinear Extension. In: Ning, L., Chau, V., Lau, F. (eds) Parallel Architectures, Algorithms and Programming. PAAP 2020. Communications in Computer and Information Science, vol 1362. Springer, Singapore. https://doi.org/10.1007/978-981-16-0010-4_8

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  • DOI: https://doi.org/10.1007/978-981-16-0010-4_8

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  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-16-0009-8

  • Online ISBN: 978-981-16-0010-4

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