Advice Complexity for a Class of Online Problems

Joan Boyar, Lene Monrad Favrholdt, Christian Kudahl, Jesper With Mikkelsen

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The advice complexity of an online problem is a measure of how much knowledge of the future an online algorithm needs in order to achieve a certain competitive ratio. We determine the advice complexity of a number of hard online problems including independent set, vertex cover, dominating set and several others. These problems are hard, since a single wrong answer by the online algorithm can have devastating consequences. For each of these problems, we show that log (1 + (c-1) c-1/c c n = Θ(n/c) bits of advice are necessary and sufficient (up to an additive term of O(log n)) to achieve a competitive ratio of c. This is done by introducing a new string guessing problem related to those of Emek et al. (TCS 2011) and Böckenhauer et al. (TCS 2014). It turns out that this gives a powerful but easy-to-use method for providing both upper and lower bounds on the advice complexity of an entire class of online problems. Previous results of Halldórsson et al. (TCS 2002) on online independent set, in a related model, imply that the advice complexity of the problem is Θ(n/c). Our results improve on this by providing an exact formula for the higher-order term. Böckenhauer et al. (ISAAC 2009) gave a lower bound of Ω(n/c) and an upper bound of O((n log c)/c) on the advice complexity of online disjoint path allocation. We improve on the upper bound by a factor of log c. For the remaining problems, no bounds on their advice complexity were previously known.

Original languageEnglish
Title of host publication32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)
EditorsErnst W. Mayr, Nicolas Ollinger
PublisherSchloss Dagstuhl-Leibniz-Zentrum fuer Informatik
Publication date2015
ISBN (Electronic)978-3-939897-78-1
Publication statusPublished - 2015
Event32nd International Symposium on Theoretical Aspects of Computer Science - Technische Universität München, Garching, Germany
Duration: 4. Mar 20157. Mar 2015


Conference32nd International Symposium on Theoretical Aspects of Computer Science
LocationTechnische Universität München
SeriesLeibniz International Proceedings in Informatics


  • Advice complexity
  • Advice complexity class AOC
  • Asymmetric string guessing
  • Covering designs
  • Online algorithms

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