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Streaming Algorithms for Monotone DR-Submodular Maximization Under a Knapsack Constraint on the Integer Lattice

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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

Maximization of non-negative monotone submodular set functions under a knapsack constraint have been extensively studied in the last decade. Here, we consider the streaming algorithms of this problem on the integer lattice, or on a multi-set equivalently. This is more realistic for many practical problems such as sensor location and influence maximization. It is well known that submodularity and diminishing return submodularity are not equivalent on the integer lattice. We mainly focus on maximizing the diminishing return submodular (DR-submodular) functions with knapsack constraint on the integer lattice. Finally, by utilizing the binary search algorithm as a subroutine, we design an online streaming algorithm called DynamicMRT. Furthermore, we prove that it is a \((1/3-\varepsilon )\)-approximation algorithm with \(O(K\log K)\) memory complexity and \(O(\log K)\) query complexity per element.

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Acknowledgements

The first author is supported by Natural Science Foundation of Shandong Province (Nos. ZR2017LA002, ZR2019MA022), and Doctoral research foundation of Weifang University (No. 2017BS02). The second author is supported by National Natural Science Foundation of China (No. 11871081). The fourth author is supported by National Natural Science Foundation of China (No. 12001025) and Science and Technology Program of Beijing Education Commission (No. KM201810005006).

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Authors and Affiliations

  1. School of Mathematics and Information Science, Weifang University, Weifang, 261061, People’s Republic of China

    Jingjing Tan

  2. School of Computer Science and Technology, Shandong Jianzhu University, Jinan, 250101, People’s Republic of China

    Dongmei Zhang

  3. Department of Operations Research and Information Engineering, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Hongyang Zhang & Zhenning Zhang

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  1. Jingjing Tan
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  2. Dongmei Zhang
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  3. Hongyang Zhang
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  4. Zhenning Zhang
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Correspondence to Dongmei Zhang .

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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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Tan, J., Zhang, D., Zhang, H., Zhang, Z. (2021). Streaming Algorithms for Monotone DR-Submodular Maximization Under a Knapsack Constraint on the Integer Lattice. 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_6

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  • DOI: https://doi.org/10.1007/978-981-16-0010-4_6

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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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