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A primal-dual approximation algorithm for the k-prize-collecting minimum power cover problem

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Abstract

In this paper, we introduce the k-prize-collecting minimum power cover problem (k-PCPC). In this problem, we are given a point set V, a sensor set S on a plane and a parameter k with \(k\le |V|\). Each sensor can adjust its power and the covering range of sensor s with power p(D(s, r(s))) is a disk D(s, r(s)), where r(s) is the radius of disk D(s, r(s)) and \(p(D(s,r(s)))=c\cdot r(s)^{\alpha }\). The k-PCPC determines a disk set \(\mathcal {F}\) such that at least k points are covered, where the center of any disk in \(\mathcal {F}\) is a sensor. The objective is to minimize the total power of the disk set \(\mathcal {F}\) plus the penalty of R, where R is the set of points that are not covered by \(\mathcal {F}\). This problem generalizes the well-known minimum power cover problem, minimum power partial cover problem and prize collecting minimum power cover problem. Our main result is to present a novel two-phase primal-dual algorithm for the k-PCPC with an approximation ratio of at most \(3^{\alpha }\).

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References

  1. Alt, H., Arkin, E.M., Brönnimann, H., Erickson, J., Fekete, S.P., Knauer, C., Lenchner, J., Mitchell, J.S.B., Whittlesey, K.: Minimum-cost coverage of point sets by disks. In: Proceedings of 22nd ACM Symposium on Computational Geometry, pp. 449–458. ACM (2006)

  2. Bar-Yehuda, R., Rawitz, D.: A note on multicovering with disk. Comput. Geom. 46(3), 394–399 (2013)

    Article  MathSciNet  Google Scholar 

  3. Bhowmick, S., Inamdar, T., Varadarajan, K.: On metric multi-covering problems. arXiv:org/abs/1602.04152 (2017)

  4. Bhowmick, S., Varadarajan, K., Xue, S.: A constant-factor approximation for multi-covering with disks. J. Comput. Geom. 6(1), 220–234 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Biló, V., Caragiannis, I., Kaklamanis, C., Kanellopoulos, P.: Geometric clustering to minimize the sum of cluster sizes. In: Proceedings of 13th annual European symposium on algorithm, pp. 460–471. Springer (2005)

  6. Charikar, M., Panigrahy, R.: Clustering to minimize the sum of cluster diameters. J. Comput. Syst. Sci. 68(2), 417–441 (2004)

    Article  MathSciNet  Google Scholar 

  7. Chvatal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4(3), 233–235 (1979)

    Article  MathSciNet  Google Scholar 

  8. Freund, A., Rawitz, D.: Combinatorial interpretations of dual fitting and primal fitting. In: Proceedings of approximation and online algorithms, pp. 137–150. Springer (2003)

  9. Guan, L., Li, W., Xiao, M.: Online algorithms for the mixed ring loading problem with two nodes. Optim. Lett. 15, 1229–1239 (2021)

    Article  MathSciNet  Google Scholar 

  10. Han, L., Xu, D., Du, D., Wu, C.: A 5-approximation algorithm for the \(k\)-prize-collecting Steiner tree problem. Optim. Lett. 13, 573–585 (2017)

    Article  MathSciNet  Google Scholar 

  11. Hou, X., Liu, W., Hou, B.: An approximation algorithm for the \(k\)-prize-collecting multicut on a tree problem. Theor. Comput. Sci. 844, 26–33 (2020)

    Article  MathSciNet  Google Scholar 

  12. Huang, Z., Feng, Q., Wang, J., Xu, J.: PTAS for minimum cost multi-covering with disks. In: Proceedings of the 2021 ACM-SIAM symposium on discrete algorithms, pp. 840–859. SIAM (2021)

  13. Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9(3), 256–278 (1974)

    Article  MathSciNet  Google Scholar 

  14. Li, M., Ran, Y., Zhang, Z.: A primal-dual algorithm for the minimum power partial cover problem. J. Comb. Optim. 39(3), 1–22 (2020)

    MathSciNet  Google Scholar 

  15. Li, M., Yang, Z., Liu, Y.: Sea depth measurement with restricted floating sensors. ACM Trans. Embed. Comput. Syst. 13(1), 1–21 (2013)

    Article  Google Scholar 

  16. Matsuda, Y., Takahashi, S.: A 4-approximation algorithm for \(k\)-prize collecting Steiner tree problems. Optim. Lett. 13(2), 341–348 (2019)

    Article  MathSciNet  Google Scholar 

  17. Pedrosa, L.L.C., Rosado, H.H.K.: A 2-approximation for the \(k\)-prize-collecting Steiner tree problem. arXiv:1911.09221 (2019)

  18. Ran, Y., Huang, X., Zhang, Z., Du, D.: Approximation algorithm for minimum power partial multi-coverage in wireless sensor networks. J. Glob. Optim. 80, 661–677 (2021)

    Article  MathSciNet  Google Scholar 

  19. Ran, Y., Shi, Y., Zhang, Z.: Local ratio method on partial set multicover. J. Comb. Optim. 34, 302–313 (2017)

    Article  MathSciNet  Google Scholar 

  20. Ran, Y., Shi, Y., Tang, C., Zhang, Z.: A primal-dual algorithm for the minimum partial set multi-cover problem. J. Comb. Optim. 39, 725–746 (2020)

    Article  MathSciNet  Google Scholar 

  21. Ran, Y., Zhang, Z., Du, H., Zhu, Y.: Approximation algorithm for partial positive influence problem in social network. J. Comb. Optim. 33, 791–802 (2017)

    Article  MathSciNet  Google Scholar 

  22. Ran, Y., Zhang, Z., Tang, S., Du, D.: Breaking the \(r_{max}\) Barrier: enhanced approximation algorithms for partial set multicover problem. INFORMS J. Comput. 33(2), 774–784 (2021)

    MathSciNet  MATH  Google Scholar 

  23. Shi, Y., Ran, Y., Zhang, Z., Du, D.: A bicriteria algorithm for the minimum submodular cost partial set multi-cover problem. Theor. Comput. Sci. 803, 1–9 (2020)

    Article  MathSciNet  Google Scholar 

  24. Shi, Y., Ran, Y., Zhang, Z., Willson, J., Tong, G., Du, D.: Approximation algorithm for the partial set multi-cover problem. J. Glob. Optim. 75, 1133–1146 (2019)

    Article  MathSciNet  Google Scholar 

  25. Slavík, P.: Improved performance of the greedy algorithm for partial cover. Inf. Process. Lett. 64(5), 251–254 (1997)

    Article  MathSciNet  Google Scholar 

  26. Vazirani, V.V.: Approximation Algorithms. Springer, Berlin (2001)

    MATH  Google Scholar 

  27. Xing, G., Lu, C., Zhang, Y., Huang, Q., Pless R.: Minimum power configuration in wireless sensor networks. In: Proceedings of the 6th ACM interational symposium on mobile ad hoc networking and computing, pp. 390–401. ACM (2005)

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Acknowledgements

The work is supported in part by the National Natural Science Foundation of China [No. 12071417], Program for Excellent Young Talents of Yunnan University, Training Program of National Science Fund for Distinguished Young Scholars, and IRTSTYN.

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Authors and Affiliations

  1. School of Information Science and Engineering, Yunnan University, Kunming, China

    Xiaofei Liu

  2. School of EECS, Peking University, Beijing, China

    Xiaofei Liu

  3. School of Mathematics and Statistics, Yunnan University, Kunming, China

    Weidong Li & Runtao Xie

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  1. Xiaofei Liu
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  2. Weidong Li
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  3. Runtao Xie
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Correspondence to Weidong Li.

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Liu, X., Li, W. & Xie, R. A primal-dual approximation algorithm for the k-prize-collecting minimum power cover problem. Optim Lett 16, 2373–2385 (2022). https://doi.org/10.1007/s11590-021-01831-z

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  • DOI: https://doi.org/10.1007/s11590-021-01831-z

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