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Approximation Algorithms for the Submodular Load Balancing with Submodular Penalties

Author

Listed:
  • Xiaofei Liu

    (School of Electronic Engineering and Computer Science, Peking University, Beijing 100871, China)

  • Peiyin Xing

    (School of Electronic Engineering and Computer Science, Peking University, Beijing 100871, China)

  • Weidong Li

    (School of Mathematics and Statistics, Yunnan University, Kunming 650504, China)

Abstract

In this paper, we study the submodular load balancing problem with submodular penalties. The objective of this problem is to balance the load among sets, while some elements can be rejected by paying some penalties. Officially, given an element set V , we want to find a subset R of rejected elements, and assign other elements to one of m sets A 1 , A 2 , ⋯ , A m . The objective is to minimize the sum of the maximum load among A 1 , A 2 , ⋯ , A m and the rejection penalty of R , where the load and rejection penalty are determined by different submodular functions. We study the submodular load balancing problem with submodular penalties under two settings: heterogenous setting (load functions are not identical) and homogenous setting (load functions are identical). Moreover, we design a Lovász rounding algorithm achieving a worst-case guarantee of m + 1 under the heterogenous setting and a min { m , ⌈ n m ⌉ + 1 } = O ( n ) -approximation combinatorial algorithm under the homogenous setting.

Suggested Citation

  • Xiaofei Liu & Peiyin Xing & Weidong Li, 2020. "Approximation Algorithms for the Submodular Load Balancing with Submodular Penalties," Mathematics, MDPI, vol. 8(10), pages 1-12, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1785-:d:428414
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    References listed on IDEAS

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    1. Xianzhao Zhang & Dachuan Xu & Donglei Du & Chenchen Wu, 2018. "Approximation algorithms for precedence-constrained identical machine scheduling with rejection," Journal of Combinatorial Optimization, Springer, vol. 35(1), pages 318-330, January.
    2. Xueling Zhong & Jinwen Ou, 2017. "Improved approximation algorithms for parallel machine scheduling with release dates and job rejection," 4OR, Springer, vol. 15(4), pages 387-406, December.
    3. Klaus Jansen & Lorant Porkolab, 2001. "Improved Approximation Schemes for Scheduling Unrelated Parallel Machines," Mathematics of Operations Research, INFORMS, vol. 26(2), pages 324-338, May.
    4. Xiaofei Liu & Weidong Li, 0. "Combinatorial approximation algorithms for the submodular multicut problem in trees with submodular penalties," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-13.
    5. Ou, Jinwen & Zhong, Xueling & Wang, Guoqing, 2015. "An improved heuristic for parallel machine scheduling with rejection," European Journal of Operational Research, Elsevier, vol. 241(3), pages 653-661.
    Full references (including those not matched with items on IDEAS)

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