Abstract
In the leasing variant of Set Cover presented by Anthony et al. [1], elements \(U\) arrive over time and must be covered by sets from a family \(F\) of subsets of \(U\). Each set can be leased for \(K\) different periods of time. Let \(\left| U \right| = n\) and \(\left| F \right| = m\). Leasing a set \(S\) for a period \(k\) incurs a cost \(c_{S}^{k}\) and allows \(S\) to cover its elements for the next \(l_k\) time steps. The objective is to minimize the total cost of the sets leased, such that elements arriving at any time \(t\) are covered by sets which contain them and are leased during time \(t\). Anthony et al. [1] gave an optimal \(O(\log n)\)-approximation for the problem in the offline setting, unless \(\mathcal {P} = \mathcal {NP}\) [22]. In this paper, we give randomized algorithms for variants of Set Cover Leasing in the online setting, including a generalization of Online Set Cover with Repetitions presented by Alon et al. [2], where elements appear multiple times and must be covered by a different set at each arrival. Our results improve the \(\mathcal {O}(\log ^2 (mn))\) competitive factor of Online Set Cover with Repetitions [2] to \(\mathcal {O}(\log d \log (dn)) = \mathcal {O}(\log m \log (mn))\), where \(d\) is the maximum number of sets an element belongs to.
This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Centre “On-The-Fly Computing” (SFB 901) and the International Graduate School “Dynamic Intelligent Systems”.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Anthony, B.M., Gupta, A.: Infrastructure leasing problems. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 424–438. Springer, Heidelberg (2007)
Alon, N., Azar, Y., Gutner, S.: Admission control to minimize rejections and online set cover with repetitions. In: 17th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), Prague, pp. 238–244 (2005)
Berman, P., DasGubta, B.: Approximating the online set multicover problems via randomized winnowing. Theor. Comput. Sci. 393(13), 54–71 (2008)
Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N.: A general approach to online network optimization problems. In: 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), New Orleans, LA, pp. 577–586 (2004)
Kling, P., Meyer auf der Heide, F., Pietrzyk, P.: An algorithm for online facility leasing. In: Even, G., Halldórsson, M.M. (eds.) SIROCCO 2012. LNCS, vol. 7355, pp. 61–72. Springer, Heidelberg (2012)
Nagarajan, C., Williamson, D.P.: Offline and online facility leasing. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 303–315. Springer, Heidelberg (2008)
Chavatal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4(3), 233–235 (1979)
Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)
Feige, U., Korman, S.: Personal communication
Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)
Lovasz, L.: On the ratio of optimal and fractional covers. Discrete Math. 13, 383–390 (1975)
Berman, P., DasGupta, B., Sontag, E.: Randomized approximation algorithms for set multicover problems with applications to reverse engineering of protein and gene networks. Discrete Appl. Math. 155(67), 733–749 (2007)
Vazirani, V.: Approximation Algorithms (2001)
Fotakis, D.: A primal-dual algorithm for online non-uniform facility location. J. Discrete Algorithms 5, 141–148 (2007)
Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.: The online set cover problem. In: 35th Annual ACM Symposium on the Theory of Computation (STOC), San Diego, CA, USA, pp. 100–105 (2003)
Awerbuch, B., Azar, Y., Fiat, A., Leighton, T.: Making commitments in the face of uncertainty: how to pick a winner almost every time. In: 28th Annual ACM Symposium on Theory of Computing (STOC), Pennsylvania, USA, pp. 519–530 (1996)
Meyerson, A.: Online facility location. In: 42nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), Las Vegas, Nevada, USA, pp. 426–431 (2001)
Fotakis, D.: On the competitive ratio for online facility location. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 637–652. Springer, Heidelberg (2003)
Buchbinder, N., Naor, J.S.: 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)
Meyerson, A.: The parking permit problem. In: 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS), Pittsburgh, PA, USA, pp. 274–284 (2005)
Malik, S., Huet, F.: Virtual cloud: rent out the rented resources. In: 6th IEEE International Conference for Internet Technology and Secured Transactions (ICITST), Abu Dhabi, UAE, pp. 536–541 (2011)
Alon, N., Moshkovitz, D., Safra, S.: Algorithmic construction of sets for k-restrictions. ACM Trans. Algorithms 2, 153–177 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Abshoff, S., Markarian, C., Meyer auf der Heide, F. (2014). Randomized Online Algorithms for Set Cover Leasing Problems. In: Zhang, Z., Wu, L., Xu, W., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2014. Lecture Notes in Computer Science(), vol 8881. Springer, Cham. https://doi.org/10.1007/978-3-319-12691-3_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-12691-3_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12690-6
Online ISBN: 978-3-319-12691-3
eBook Packages: Computer ScienceComputer Science (R0)