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Maximizing a monotone non-submodular function under a knapsack constraint

Author

Listed:
  • Zhenning Zhang

    (Beijing University of Technology)

  • Bin Liu

    (Ocean University of China)

  • Yishui Wang

    (Chinese Academy of Sciences)

  • Dachuan Xu

    (Beijing University of Technology)

  • Dongmei Zhang

    (Shandong Jianzhu University)

Abstract

Submodular optimization has been well studied in combinatorial optimization. However, there are few works considering about non-submodular optimization problems which also have many applications, such as experimental design, some optimization problems in social networks, etc. In this paper, we consider the maximization of non-submodular function under a knapsack constraint, and explore the performance of the greedy algorithm, which is characterized by the submodularity ratio $$\beta $$ β and curvature $$\alpha $$ α . In particular, we prove that the greedy algorithm enjoys a tight approximation guarantee of $$ (1-e^{-\alpha \beta })/{\alpha }$$ ( 1 - e - α β ) / α for the above problem. To our knowledge, it is the first tight constant factor for this problem. We further utilize illustrative examples to demonstrate the performance of our algorithm.

Suggested Citation

  • Zhenning Zhang & Bin Liu & Yishui Wang & Dachuan Xu & Dongmei Zhang, 0. "Maximizing a monotone non-submodular function under a knapsack constraint," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-24.
    • Handle: RePEc:spr:jcomop:v::y::i::d:10.1007_s10878-020-00620-1
    DOI: 10.1007/s10878-020-00620-1
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    References listed on IDEAS

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    1. Fisher, M.L. & Nemhauser, G.L. & Wolsey, L.A., 1978. "An analysis of approximations for maximizing submodular set functions," LIDAM Reprints CORE 341, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Fisher, M.L. & Nemhauser, G.L. & Wolsey, L.A., 1978. "An analysis of approximations for maximizing submodular set functions - 1," LIDAM Reprints CORE 334, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Akiyoshi Shioura & Natalia V. Shakhlevich & Vitaly A. Strusevich, 2016. "Application of Submodular Optimization to Single Machine Scheduling with Controllable Processing Times Subject to Release Dates and Deadlines," INFORMS Journal on Computing, INFORMS, vol. 28(1), pages 148-161, February.
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