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. Using advice complexity, we define the first online complexity class, AOC. The class includes independent set, vertex cover, dominating set, and several others as complete problems. The complete problems in this class 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))/cc)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.
The results are obtained by introducing a new string guessing problem related to those of Emek et al. (TCS 2011) and B¨ockenhauer 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, the AOCcomplete 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. For online disjoint path allocation, 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. 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.