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Theory of Computing Systems

, Volume 62, Issue 6, pp 1443–1469 | Cite as

Weighted Online Problems with Advice

  • Joan Boyar
  • Lene M. Favrholdt
  • Christian Kudahl
  • Jesper W. Mikkelsen
Article
  • 58 Downloads
Part of the following topical collections:
  1. Special Issue on Combinatorial Algorithms

Abstract

Recently, the first online complexity class, A O C, was introduced. The class consists of many online problems where each request must be either accepted or rejected, and the aim is to either minimize or maximize the number of accepted requests, while maintaining a feasible solution. All A O C-complete problems (including Independent Set, Vertex Cover, Dominating Set, and Set Cover) have essentially the same advice complexity. In this paper, we study weighted versions of problems in A O C, i.e., each request comes with a weight and the aim is to either minimize or maximize the total weight of the accepted requests. In contrast to the unweighted versions, we show that there is a significant difference in the advice complexity of complete minimization and maximization problems. We also show that our algorithmic techniques for dealing with weighted requests can be extended to work for non-complete A O C problems such as Matching in the edge arrival model (giving better results than what follow from the general A O C results) and even non- A O C problems such as scheduling.

Keywords

Online algorithms Weighted online problems Advice complexity AOC Scheduling 

Notes

Acknowledgments

This work was partially supported by the Villum Foundation, grant VKR023219, and the Danish Council for Independent Research, Natural Sciences, grant DFF-1323-00247.

References

  1. 1.
    Albers, S., Hellwig, M.: Online Makespan Minimization with Parallel Schedules. In: SWAT, LNCS, vol. 8503, pp 13–25 (2014)Google Scholar
  2. 2.
    Böckenhauer, H. J., Hromkovič, J., Komm, D., Krug, S., Smula, J., Sprock, A.: The string guessing problem as a method to prove lower bounds on the advice complexity. Theor. Comput. Sci. 554, 95–108 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Böckenhauer, H. J., Komm, D., Královič, R., Královič, R., Mömke, T.: On the Advice Complexity of Online Problems. In: ISAAC, LNCS, vol. 5878, pp 331–340 (2009)Google Scholar
  4. 4.
    Borodin, A., Irani, S., Raghavan, P., Schieber, B.: Competitive paging with locality of reference. J. Comput. Syst. Sci. 50(2), 244–258 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boyar, J., Epstein, L., Favrholdt, L.M., Larsen, K.S., Levin, A.: Online Bounded Analysis. In: CSR, LNCS, vol. 9691, pp 131–145 (2016)Google Scholar
  6. 6.
    Boyar, J., Favrholdt, L.M., Kudahl, C., Larsen, K.S., Mikkelsen, J.W.: Online algorithms with advice: a survey. ACM Comput. Surv. (CSUR) 50(2), 19 (2017)CrossRefzbMATHGoogle Scholar
  7. 7.
    Boyar, J., Favrholdt, L.M., Kudahl, C., Mikkelsen, J.W.: Advice Complexity for a Class of Online Problems. In: STACS, LIPIcs, vol. 30, pp. 116–129 (2015). Full paper to appear in Theory of Computing SystemsGoogle Scholar
  8. 8.
    Chrobak, M., Noga, J.: LRU is better than FIFO. Algorithmica 23(2), 180–185 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dobrev, S., Královič, R., Pardubská, D.: Measuring the problem-relevant information in input. RAIRO - Theor. Inf. Appl. 43(3), 585–613 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dürr, C., Konrad, C., Renault, M.P.: On the Power of Advice and Randomization for Online Bipartite Matching. In: ESA, LIPIcs, pp 37:1–37:16 (2016)Google Scholar
  11. 11.
    Emek, Y., Fraigniaud, P., Korman, A., Rosén, A.: Online computation with advice. Theor. Comput. Sci. 412(24), 2642–2656 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Karlin, A.R., Manasse, M.S., Rudolph, L., Sleator, D.D.: Competitive snoopy caching. Algorithmica 3, 77–119 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Komm, D., Královič, R., Královič, R., Kudahl, C.: Advice Complexity of the Online Induced Subgraph Problem. In: MFCS, LIPIcs, vol. 58, pp 59:1–59:13 (2016)Google Scholar
  14. 14.
    Mikkelsen, J.W.: Randomization can be as Helpful as a Glimpse of the Future in Online Computation. In: ICALP, LIPIcs, vol. 55, pp 39:1–39:14 (2016)Google Scholar
  15. 15.
    Renault, M.P., Rosén, A., van Stee, R.: Online algorithms with advice for bin packing and scheduling problems. Theor. Comput. Sci. 600, 155–170 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Commun. ACM 28(2), 202–208 (1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Smula, J.: Information Content of Online Problems: Advice Versus Determinism and Randomization. Ph.D. Thesis, ETH Zürich (2015)Google Scholar
  18. 18.
    Sprock, A.: Analysis of Hard Problems in Reoptimization and Online Computation. Ph.D. thesis, ETH Zürich (2013)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdenseDenmark

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