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New Bounds for Online Packing LPs

  • Matthias Englert
  • Nicolaos Matsakis
  • Marcin Mucha
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)

Abstract

Solving linear programs online has been an active area of research in recent years and was used with great success to develop new online algorithms for a variety of problems. We study the setting introduced by Ochel et al. as an abstraction of lifetime optimization of wireless sensor networks.

In this setting, the online algorithm is given a packing LP and has to monotonically increase LP variables in order to maximize the objective function. However, at any point in time, the adversary only provides an α-approximation of the remaining slack for each constraint. This is designed to model scenarios in which only estimates of remaining capacities (e.g. of batteries) are known, and they get more and more accurate as the remaining capacities approach 0.

Ochel et al. (ICALP’12) gave a Θ(ln α/α)-competitive online algorithm for this online packing LP problem and showed an upper bound on the competitive ratio of any online algorithm, even randomized, of \(O(1/\sqrt{\alpha})\). We significantly improve the upper bound and show that any deterministic online algorithm for LPs with d variables is at most O(d 2 α 1/d /α)-competitive. For randomized online algorithms we show an upper bound of O(m 2 α 1/m /α) for LPs with m m!ln α variables. For LPs with sufficiently many variables, these bounds are O(ln 2 α/α), nearly matching the known lower bound.

On the other hand, we also show that the known lower bound can be significantly improved if the number of variables in the LP is small. Specifically, we give a deterministic \(\Theta(1/\sqrt{\alpha})\)-competitive online algorithm for packing LPs with two variables. This is tight, since the previously known upper bound of \(O(1/\sqrt{\alpha})\) still holds for 2-dimensional LPs.

Keywords

Wireless Sensor Network Network Lifetime Competitive Ratio Online Algorithm Deterministic 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.

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References

  1. 1.
    Adamaszek, A., Czumaj, A., Englert, M., Räcke, H.: An O(logk)-competitive algorithm for generalized caching. In: Proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1681–1689 (2012)Google Scholar
  2. 2.
    Bansal, N., Buchbinder, N., Madry, A., Naor, J.: A polylogarithmic-competitive algorithm for the k-server problem. In: Proceedings of the 52nd IEEE Symposium on Foundations of Computer Science (FOCS), pp. 267–276 (2011)Google Scholar
  3. 3.
    Buchbinder, N., Naor, J.: Online primal-dual algorithms for covering and packing. Mathematics of Operations Research 34(2), 270–286 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Calinescu, G., Kapoor, S., Olshevsky, A., Zelikovsky, A.: Network lifetime and power assignment in ad hoc wireless networks. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 114–126. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. 5.
    Ochel, M., Radke, K., Vöcking, B.: Online packing with gradually improving capacity estimations and applications to network lifetime maximization. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part II. LNCS, vol. 7392, pp. 648–659. Springer, Heidelberg (2012)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Matthias Englert
    • 1
  • Nicolaos Matsakis
    • 1
  • Marcin Mucha
    • 2
  1. 1.DIMAP and Dept. of Computer ScienceUniversity of WarwickUK
  2. 2.Institute of InformaticsUniversity of WarsawPoland

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