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Greedy Algorithm for Maximization of Non-submodular Functions Subject to Knapsack Constraint

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Computing and Combinatorics (COCOON 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11653))

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Abstract

Although submodular maximization generalizes many fundamental problems in discrete optimization, lots of real-world problems are non-submodular. In this paper, we consider the maximization problem of non-submodular function with a knapsack constraint, and explore the performance of the greedy algorithm. Our guarantee is characterized by the submodularity ratio \(\beta \) and curvature \(\alpha \). In particular, we prove that the greedy algorithm enjoys a tight approximation guarantee of \(\frac{1}{\alpha }\left( 1-e^{-\alpha \beta }\right) \) for the above problem. To our knowledge, it is the first tight constant factor for this problem. In addition, we experimentally validate our algorithm by an important application, the Bayesian A-optimality.

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Acknowledgment

The first author is supported by Science and Technology Program of Beijing Education Commission (No. KM201810005006). The second author is supported by Natural Science Foundation of Shandong Province of China (No. ZR2017QA010). The third author is supported by National Natural Science Foundation of China (No. 61433012), Shenzhen research grant (KQJSCX20180330170311901, JCYJ20180305180840138 and GGFW2017073114031767). The fourth author is supported by National Natural Science Foundation of China (No. 11531014). The fifth author is supported by National Natural Science Foundation of China (No. 11871081).

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Authors and Affiliations

  1. College of Applied Sciences, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Zhenning Zhang

  2. School of Mathematical Science, Ocean University of China, Qingdao, 266100, People’s Republic of China

    Bin Liu

  3. Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, 518055, People’s Republic of China

    Yishui Wang

  4. Department of Operations Research and Scientific Computing, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Dachuan Xu

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

    Dongmei Zhang

Authors
  1. Zhenning Zhang
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  2. Bin Liu
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  3. Yishui Wang
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  4. Dachuan Xu
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  5. Dongmei Zhang
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Corresponding author

Correspondence to Yishui Wang .

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Editors and Affiliations

  1. The University of Texas at Dallas, Richardson, TX, USA

    Ding-Zhu Du

  2. Xidian University, Xi'an, China

    Zhenhua Duan

  3. Xidian University, Xi'an, China

    Cong Tian

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Zhang, Z., Liu, B., Wang, Y., Xu, D., Zhang, D. (2019). Greedy Algorithm for Maximization of Non-submodular Functions Subject to Knapsack Constraint. In: Du, DZ., Duan, Z., Tian, C. (eds) Computing and Combinatorics. COCOON 2019. Lecture Notes in Computer Science(), vol 11653. Springer, Cham. https://doi.org/10.1007/978-3-030-26176-4_54

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  • DOI: https://doi.org/10.1007/978-3-030-26176-4_54

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26175-7

  • Online ISBN: 978-3-030-26176-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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