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Breaking the r max Barrier: Enhanced Approximation Algorithms for Partial Set Multicover Problem

Author

Listed:
  • Yingli Ran

    (College of Mathematics and Computer Sciences, Zhejiang Normal University, Jinhua, Zhejiang 321004, China)

  • Zhao Zhang

    (College of Mathematics and Computer Sciences, Zhejiang Normal University, Jinhua, Zhejiang 321004, China)

  • Shaojie Tang

    (Naveen Jindal School of Management, University of Texas at Dallas, Richardson, Texas 75080)

  • Ding-Zhu Du

    (Department of Computer Science, University of Texas at Dallas, Richardson, Texas 75080)

Abstract

Given an element set E of order n , a collection of subsets S ⊆ 2 E , a cost c S on each set S ∈ S , a covering requirement r e for each element e ∈ E , and an integer k , the goal of a minimum partial set multicover problem (MinPSMC) is to find a subcollection F ⊆ S to fully cover at least k elements such that the cost of F is as small as possible and element e is fully covered by F if it belongs to at least r e sets of F . This problem generalizes the minimum k -union problem (Min k U) and is believed not to admit a subpolynomial approximation ratio. In this paper, we present a ( 4 log ⁡ n H ( Δ ) In k + 2 ⁡ log n n ) -approximation algorithm for MinPSMC, in which Δ is the maximum size of a set in S . And when k = Ω ( n ) , we present a bicriteria algorithm fully covering at least ( 1 − ε 2 log n ) k elements with approximation ratio O ( 1 ε ( log n ) 2 H ( Δ ) ) , where 0 < ε < 1 is a fixed number. These results are obtained by studying the minimum density subcollection problem with (or without) cardinality constraint, which might be of interest by itself.

Suggested Citation

  • Yingli Ran & Zhao Zhang & Shaojie Tang & Ding-Zhu Du, 2021. "Breaking the r max Barrier: Enhanced Approximation Algorithms for Partial Set Multicover Problem," INFORMS Journal on Computing, INFORMS, vol. 33(2), pages 774-784, May.
    • Handle: RePEc:inm:orijoc:v:33:y:2021:i:2:p:774-784
    DOI: 10.1287/ijoc.2020.0975
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    References listed on IDEAS

    as
    1. Thang N. Dinh & Yilin Shen & Dung T. Nguyen & My T. Thai, 2014. "On the approximability of positive influence dominating set in social networks," Journal of Combinatorial Optimization, Springer, vol. 27(3), pages 487-503, April.
    2. Yishuo Shi & Yingli Ran & Zhao Zhang & James Willson & Guangmo Tong & Ding-Zhu Du, 2019. "Approximation algorithm for the partial set multi-cover problem," Journal of Global Optimization, Springer, vol. 75(4), pages 1133-1146, December.
    3. Yingli Ran & Yishuo Shi & Changbing Tang & Zhao Zhang, 2020. "A primal-dual algorithm for the minimum partial set multi-cover problem," Journal of Combinatorial Optimization, Springer, vol. 39(3), pages 725-746, April.
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    Citations

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    Cited by:

    1. Zhao Zhang & Wei Liang & Hongmin W. Du & Siwen Liu, 2022. "Constant Approximation for the Lifetime Scheduling Problem of p -Percent Coverage," INFORMS Journal on Computing, INFORMS, vol. 34(5), pages 2675-2685, September.

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