Abstract
Given a ground set \(\mathcal {U}\) of n elements and a family of m subsets \(\mathcal {S} = \{S_i : S_i\subseteq \mathcal {U}\}\). Each subset \(S\in \mathcal {S}\) has a positive cost c(S) and every element \(e\in \mathcal {U}\) is associated with an integer coverage requirement \(r_e>0\), which means that e has to be covered at least \(r_e\) times. The weighted set multi-cover problem asks for the minimum cost subcollection which covers all of the elements such that each element e is covered at least \(r_e\) times.
In this paper, we study the online version of the weighted set multi-cover problem. We give a randomized algorithm with competitive ratio \(8(1+\ln m)\ln n\) for this problem based on the primal-dual method, which improve previous competitive ratio \(12\log m\log n\) for the online set multi-cover problem that is the special version where each cost c(S) is 1 for every subset S.
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References
Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.: The online set cover problem. In: STOC 2003, pp. 100–105 (2003)
Bansal, N., Buchbinder, N., Naor, J.: A primal-dual randomized algorithm for weighted paging. In: FOCS 2007, pp. 507–517 (2007)
Buchbinder, N., Jain, K., Naor, J.S.: Online primal-dual algorithms for maximizing ad-auctions revenue. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 253–264. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-75520-3_24
Buchbinder, N., Naor, J.: Online primal-dual algorithms for covering and packing problems. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 689–701. Springer, Heidelberg (2005). https://doi.org/10.1007/11561071_61
Buchbinder, N., Naor, J.: Improved bounds for online routing and packing via a primal-dual approach. In: Proceedings of the 47th Symposium on Foundations of Computer Science (FOCS), pp. 293–304 (2006)
Buchbinder, N., Naor, J.: The design of competitive online algorithms via a primal-dual approach. Found. Trends Theor. Comput. Sci. 3(2–3), 93–263 (2009)
Chvatal, V.: A greedy heuristic for the set covering problem. Math. Oper. Res. 4, 233–235 (1979)
Feige, U.: A threshold of ln n for approximating set cover. In: Proceedings of the 28th ACM Symposium on the Theory of Computing, pp. 312–318 (1996)
Feige, U.: A threshold of \(\ln n\) for approximating set cover. J. ACM 45(4), 634–652 (1998)
Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)
Kuhnle, A., Li, X., Smith, J.D., Thai, M.T.: Online set multicover algorithms for dynamic D2D communications. J. Comb. Optim. 34(4), 1237–1264 (2017). https://doi.org/10.1007/s10878-017-0144-y
Lov\(\acute{a}\)sz, L.: On the ratio of optimal integral and fractional covers. Discrete Math. 13, 383–390 (1975)
Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. In: Proceedings of the 25th ACM Symposium on Theory of Computing, pp. 286–293 (1993)
Rajagopalan, S., Vazirani, V.V.: Primal-dual RNC approximation algorithms for set cover and covering integer programs. SIAM J. Comput. 109(28), 525–540 (1998)
Sun, Z., Li, L., Li, X., Xing, X., Li, Y.: Optimization coverage conserving protocol with authentication in wireless sensor networks. Int. J. Distrib. Sens. Netw. 13(3), 1–16 (2017)
Sun, Z., Li, C., Xing, X., Wang, H., Yan, B., Li, X.: K-degree coverage algorithm based on optimization nodes deployment in wireless sensor networks. Int. J. Distrib. Sens. Netw. 13(2), 1–16 (2017)
Sun, Z., Shu, Y., Xing, X., et al.: LPOCS: a novel linear programming optimization coverage scheme in wireless sensor networks. J. Ad Hoc Sens. Wirel. Netw. 33(1/4), 173–197 (2016)
Sun, Z., Zhang, Y., Xing, X., et al.: EBKCCA: a novel energy balanced \(k\)-coverage control algorithm based on probability model in wireless sensor networks. KSII Trans. Internet Inf. Syst. 10(8), 3621–3640 (2016)
Sun, Z., Wang, H., Wu, W., Xing, X.: ECAPM: an enhanced coverage algorithm in wireless sensor network based on probability model. Int. J. Distrib. Sens. Netw. 2015Article ID 203502, 11 pages (2015)
Young, N.E.: The k-server dual and loose competitiveness for paging. Algorithmica 11, 525–541 (1994). Preliminary version appeared in SODA’91 titled “On-Line Caching as Cache Size Varies”
Acknowledgments
We would like to thank the anonymous referees for their careful readings of the manuscripts and many useful suggestions.
Wenbin Chen’s research has been supported by the National Science Foundation of China (NSFC) under Grant No. 11271097, and by the Project of Ordinary University Innovation Team Construction of Guangdong Province Under No. 2015KCXTD014 and No. 2016KCXTD017. This work has been also supported by the Natural Science Foundation of China (U1936116), the Guangxi Key Laboratory of Cryptography and Information Security (GCIS201807). FuFang Li’s work had been co-financed by: Natural Science Foundation of China under Grant No. 61472092; Guangdong Provincial Science and Technology Plan Project under Grant No. 2013B010401037; and GuangZhou Municipal High School Science Research Fund under grant No. 1201421317. Ke Qi’s research has been supported by the Guangzhou Science and Technology Plan Project under Grant No. 201707010283 and the National Science Foundation of Guangdong Province under Grant No. 2017A030313374. Miao Liu’s research has been supported by the Guangzhou Municipal Universities project 1201620342.
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Chen, W., Li, F., Qi, K., Liu, M., Tang, M. (2020). A Primal-Dual Randomized Algorithm for the Online Weighted Set Multi-cover Problem. In: Chen, J., Feng, Q., Xu, J. (eds) Theory and Applications of Models of Computation. TAMC 2020. Lecture Notes in Computer Science(), vol 12337. Springer, Cham. https://doi.org/10.1007/978-3-030-59267-7_6
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