An Algorithm for Online Facility Leasing

  • Peter Kling
  • Friedhelm Meyer auf der Heide
  • Peter Pietrzyk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7355)

Abstract

We consider an online facility location problem where clients arrive over time and their demands have to be served by opening facilities and assigning the clients to opened facilities. When opening a facility we must choose one of K different lease types to use. A lease type k has a certain lease length l k . Opening a facility i using lease type k causes a cost of \(f_i^k\) and ensures that i is open for the next l k time steps. In addition to costs for opening facilities, we have to take connection costs c ij into account when assigning a client j to facility i. We develop and analyze the first online algorithm for this problem that has a time-independent competitive factor.

This variant of the online facility location problem was introduced by [7] and is strongly related to both the online facility problem by [5] and the parking permit problem by [6]. Nagarajan and Williamson gave a 3-approximation algorithm for the offline problem and an O (Klogn)-competitive algorithm for the online variant. Here, n denotes the total number of clients arriving over time. We extend their result by removing the dependency on n (and thereby on the time). In general, our algorithm is \(O (\ensuremath{l_{\text{max}}} \log(\ensuremath{l_{\text{max}}} ))\)-competitive. Here \(\ensuremath{l_{\text{max}}} \) denotes the maximum lease length. Moreover, we prove that it is \(O (\log^2(\ensuremath{l_{\text{max}}} ))\)-competitive for many “natural” cases. Such cases include, for example, situations where the number of clients arriving in each time step does not vary too much, or is non-increasing, or is polynomially bounded in \(l_{\text{max}}\).

Keywords

Problem Instance Facility Location Competitive Ratio Online Algorithm Competitive Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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), doi:10.1007/978-3-540-72792-7_32, ISBN 978-3-540-72791-0CrossRefGoogle Scholar
  2. 2.
    Blelloch, G.E., Tangwongsan, K.: Parallel approximation algorithms for facility-location problems. In: Proceedings of the 22nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pp. 315–324 (2010) ISBN 978-1-4503-0079-7Google Scholar
  3. 3.
    Fotakis, D.: A primal-dual algorithm for online non-uniform facility location. Journal of Discrete Algorithms 5(1), 141–148 (2007), doi:10.1016/j.jda.2006.03.001, ISSN 1570-8667MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and lagrangian relaxation. Journal of the ACM 48(2), 274–296 (2001) ISSN 0004-5411MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Meyerson, A.: Online facility location. In: Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science (FOCS), pp. 426–431. IEEE Computer Society, Washington, DC (2001) ISBN 0-7695-1390-5Google Scholar
  6. 6.
    Meyerson, A.: The parking permit problem. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 274–284. IEEE Computer Society, Washington, DC (2005), doi:10.1109/SFCS.2005.72, ISBN 0-7695-2468-0CrossRefGoogle Scholar
  7. 7.
    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), doi:10.1007/978-3-540-68891-4_21, ISBN 978-3-540-68886-0 CrossRefGoogle Scholar
  8. 8.
    Pandit, S., Pemmaraju, S.V.: Rapid randomized pruning for fast greedy distributed algorithms. In: Proceeding of the 29th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC), pp. 325–334. ACM, New York (2010), doi:10.1145/1835698.1835777, ISBN 978-1-60558-888-9 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Peter Kling
    • 1
  • Friedhelm Meyer auf der Heide
    • 1
  • Peter Pietrzyk
    • 1
  1. 1.Heinz Nixdorf Institute & Computer Science DepartmentUniversity of PaderbornPaderbornGermany

Personalised recommendations