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A Technique to Obtain Hardness Results for Randomized Online Algorithms – A Survey

  • Hans-Joachim Böckenhauer
  • Juraj HromkovičEmail author
  • Dennis Komm
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8808)

Abstract

We survey how the advice complexity of online algorithms can be used to obtain lower bounds on the performance of randomized online algorithms. Online algorithms with advice may query an oracle that knows the whole input from the start to solve some instance of an online problem. This is done by reading a finite prefix of some infinite binary advice tape, which is created by the oracle before the first piece of input is processed. Similarly, a randomized online algorithm may use a binary tape where every bit is chosen uniformly at random.

In this survey, we review a technique, similar to Yao’s principle, which allows statements on the advice complexity of some given online problem to translate to results on the power of randomization for this problem in terms of lower bounds. We give some examples where this technique works and how it is applied, and show its limitations and that it is tight in a very general sense.

Keywords

Competitive Ratio Online Algorithm Online Computation Input Length Deterministic Strategy 
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.
    Barhum, K.: Tight Bounds for the Advice Complexity of the Online Minimum Steiner Tree Problem. In: Geffert, V., Preneel, B., Rovan, B., Štuller, J., Tjoa, A.M. (eds.) SOFSEM 2014. LNCS, vol. 8327, pp. 77–88. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  2. 2.
    Barhum, K., Böckenhauer, H.-J., Forišek, M., Gebauer, H., Hromkovič, J., Krug, S., Smula, J., Steffen, B.: On the Power of Advice and Randomization for the Disjoint Path Allocation Problem. In: Geffert, V., Preneel, B., Rovan, B., Štuller, J., Tjoa, A.M. (eds.) SOFSEM 2014. LNCS, vol. 8327, pp. 89–101. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  3. 3.
    Bianchi, M.P., Böckenhauer, H.-J., Hromkovič, J., Keller, L.: Online Coloring of Bipartite Graphs with and without Advice. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 519–530. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  4. 4.
    Bianchi, M.P., Böckenhauer, H.-J., Hromkovič, J., Krug, S., Steffen, B.: On the Advice Complexity of the Online L(2,1)-Coloring Problem on Paths and Cycles. In: Du, D.-Z., Zhang, G. (eds.) COCOON 2013. LNCS, vol. 7936, pp. 53–64. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  5. 5.
    Bansal, N., Buchbinder, N., Madry, A., Naor, J.: A polylogarithmic-competitive algorithm for the \(k\)-server problem (extended abstract). In: Proc. of FOCS 2011, pp. 267–276 (2011)Google Scholar
  6. 6.
    Böckenhauer, H.-J., Hromkovič, J., Komm, D., Královič, R., Rossmanith, P.: On the Power of Randomness versus Advice in Online Computation. In: Bordihn, H., Kutrib, M., Truthe, B. (eds.) Languages Alive. LNCS, vol. 7300, pp. 30–43. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  7. 7.
    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. In: Du, D.-Z., Zhang, G. (eds.) COCOON 2013. LNCS, vol. 7936, pp. 493–505. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  8. 8.
    Böckenhauer, H.J., Komm, D., Královič, R., Královič, R.: On the advice complexity of the \(k\)-server problem. In: Aceto, L., Henzinger, M., Sgall, J. (eds.): ICALP 2011, Part I. LNCS, vol. 6755, pp. 207–218. Springer, Heidelberg (2011)Google Scholar
  9. 9.
    Böckenhauer, H.-J., Komm, D., Královič, R., Královič, R., Mömke, T.: On the Advice Complexity of Online Problems. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 331–340. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Böckenhauer, H.-J., Komm, D., Královič, R., Rossmanith, P.: On the Advice Complexity of the Knapsack Problem. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 61–72. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  11. 11.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press (1998)Google Scholar
  12. 12.
    Boyar, J., Kamali, S., Larsen, K.S., López-Ortiz, A.: Online bin packing with advice. In: Proc. of STACS 2014. LIPIcs 25, pp. 174–186. Schloss Dagstuhl (2014)Google Scholar
  13. 13.
    Boyar, J., Kamali, S., Larsen, K.S., López-Ortiz, A.: On the List Update Problem with Advice. In: Dediu, A.-H., Martín-Vide, C., Sierra-Rodríguez, J.-L., Truthe, B. (eds.) LATA 2014. LNCS, vol. 8370, pp. 210–221. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  14. 14.
    Dorrigiv, R., He, M., Zeh, N.: On the Advice Complexity of Buffer Management. In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 136–145. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  15. 15.
    Dobrev, S., Královič, R., Královič, R.: Independent Set with Advice: The Impact of Graph Knowledge. In: Erlebach, T., Persiano, G. (eds.) WAOA 2012. LNCS, vol. 7846, pp. 2–15. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  16. 16.
    I. Seleceniova. Personal communication. Google Scholar
  17. 17.
    Dobrev, S., Královič, R., Markou, E.: Online Graph Exploration with Advice. In: Even, G., Halldórsson, M.M. (eds.) SIROCCO 2012. LNCS, vol. 7355, pp. 267–278. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  18. 18.
    Dobrev, S., Královič, R., Pardubská, D.: How Much Information about the Future Is Needed? In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds.) SOFSEM 2008. LNCS, vol. 4910, pp. 247–258. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  19. 19.
    Dohrau, J.: Online makespan scheduling with sublinear advice. Technical Report, ETH Zurich (2013)Google Scholar
  20. 20.
    Emek, Y., Fraigniaud, P., Korman, A., Rosén, A.: Online Computation with Advice. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 427–438. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  21. 21.
    Elias, P.: Universal codeword sets and representations of the integers. IEEE Transactions on Information Theory 21(2), 194–203 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Forišek, M., Keller, L., Steinová, M.: Advice Complexity of Online Coloring for Paths. In: Dediu, A.-H., Martín-Vide, C. (eds.) LATA 2012. LNCS, vol. 7183, pp. 228–239. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  23. 23.
    Gupta, S., Kamali, S., López-Ortiz, A.: On Advice Complexity of the k-server Problem under Sparse Metrics. In: Moscibroda, T., Rescigno, A.A. (eds.) SIROCCO 2013. LNCS, vol. 8179, pp. 55–67. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  24. 24.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics, 2nd ed. Addison-Wesley (1994)Google Scholar
  25. 25.
    Griggs, J.R., Yeh, R.K.: Labelling graphs with a condition at distance \(2\). SIAM Journal on Discrete Mathemtics 5(4), 586–595 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hromkovič, J., Královič, R., Královič, R.: Information Complexity of Online Problems. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 24–36. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  27. 27.
    Hromkovič, J., Mömke, T., Steinhöfel, K., Widmayer, P.: Job shop scheduling with unit length tasks: Bounds and algorithms. Algorithmic Operations Research 2(1), 1–14 (2007)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Komm, D., Královič, R.: Advice complexity and barely random algorithms. Theoretical Informatics and Applications (RAIRO) 45(2), 249–267 (2011)Google Scholar
  29. 29.
    Komm, D., Královič, R., Mömke, T.: On the Advice Complexity of the Set Cover Problem. In: Hirsch, E.A., Karhumäki, J., Lepistö, A., Prilutskii, M. (eds.) CSR 2012. LNCS, vol. 7353, pp. 241–252. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  30. 30.
    Koutsoupias, E., Papadimitriou, C.H.: On the \(k\)-server conjecture. Journal of the ACM 42(5), 971–983 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Manasse, M.S., McGeoch, L.A., Sleator, D.D.: Competitive algorithms for on-line problems. In: Proc. of STOC 1998, pp. 322–333 (1988)Google Scholar
  32. 32.
    Renault, M.P., Rosén, A.: On Online Algorithms with Advice for the k-Server Problem. In: Solis-Oba, R., Persiano, G. (eds.) WAOA 2011. LNCS, vol. 7164, pp. 198–210. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  33. 33.
    Renault, M.P., Rosén, A., van Stee, R.: Online algorithms with advice for bin packing and scheduling problems. CoRR abs/1311.7589 (2013)Google Scholar
  34. 34.
    Seibert, S., Sprock, A., Unger, W.: Advice Complexity of the Online Coloring Problem. In: Spirakis, P.G., Serna, M. (eds.) CIAC 2013. LNCS, vol. 7878, pp. 345–357. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  35. 35.
    Yao, A.C.-C.: Probabilistic computations: Toward a unified measure of complexity (extended abstract). In: Proc. of FOCS 1977, pp. 222–227. IEEE Computer Society (1977)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Hans-Joachim Böckenhauer
    • 1
  • Juraj Hromkovič
    • 1
    Email author
  • Dennis Komm
    • 1
  1. 1.Department of Computer ScienceETH ZürichZürichSwitzerland

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