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A primal-dual algorithm for the minimum partial set multi-cover problem

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Listed:
  • Yingli Ran

    (Zhejiang Normal University)

  • Yishuo Shi

    (Institute of Information Science, Academia Sinica)

  • Changbing Tang

    (Zhejiang Normal University)

  • Zhao Zhang

    (Zhejiang Normal University)

Abstract

In a minimum partial set multi-cover problem (MinPSMC), given an element set E, a collection of subsets $${\mathcal {S}} \subseteq 2^E$$S⊆2E, a cost $$w_S$$wS on each set $$S\in {\mathcal {S}}$$S∈S, a covering requirement $$r_e$$re for each element $$e\in E$$e∈E, and an integer k, the goal is to find a sub-collection $${\mathcal {F}} \subseteq {\mathcal {S}}$$F⊆S to fully cover at least k elements such that the cost of $${\mathcal {F}}$$F is as small as possible, where element e is fully covered by $${\mathcal {F}}$$F if it belongs to at least $$r_e$$re sets of $${\mathcal {F}}$$F. On the application side, the problem has its background in the seed selection problem in a social network. On the theoretical side, it is a natural combination of the minimum partial (single) set cover problem (MinPSC) and the minimum set multi-cover problem (MinSMC). Although both MinPSC and MinSMC admit good approximations whose performance ratios match those lower bounds for the classic set cover problem, previous studies show that theoretical study on MinPSMC is quite challenging. In this paper, we prove that MinPSMC cannot be approximated within factor $$O(n^\frac{1}{2(\log \log n)^c})$$O(n12(loglogn)c) for some constant c under the ETH assumption. Furthermore, assuming $$r_{\max }$$rmax is a constant, where $$r_{\max } =\max _{e\in E} r_e$$rmax=maxe∈Ere is the maximum covering requirement and f is the maximum number of sets containing a common element, we present a primal-dual algorithm for MinPSMC and show that its performance ratio is $$O(\sqrt{n})$$O(n). We also improve the ratio for a restricted version of MinPSMC which possesses a graph-type structure.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:jcomop:v:39:y:2020:i:3:d:10.1007_s10878-019-00513-y
    DOI: 10.1007/s10878-019-00513-y
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    References listed on IDEAS

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    1. V. Chvatal, 1979. "A Greedy Heuristic for the Set-Covering Problem," Mathematics of Operations Research, INFORMS, vol. 4(3), pages 233-235, August.
    2. 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.
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    Cited by:

    1. Yingli Ran & Xiaohui Huang & Zhao Zhang & Ding-Zhu Du, 2021. "Approximation algorithm for minimum power partial multi-coverage in wireless sensor networks," Journal of Global Optimization, Springer, vol. 80(3), pages 661-677, July.
    2. 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.
    3. Man Xiao & Weidong Li & Xiaofei Liu, 2023. "Online Mixed Ring Covering Problem with Two Nodes," SN Operations Research Forum, Springer, vol. 4(1), pages 1-20, March.

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