Advice Complexity of Online Coloring for Paths

In online graph coloring a graph is revealed to an online algorithm one vertex at a time, and the algorithm must color the vertices as they appear. This paper starts to investigate the advice complexity of this problem --- the amount of oracle information an online algorithm needs in order to make optimal choices. We also consider a more general problem --- a trade-off between online and offline graph coloring. In the paper we prove that precisely ⌈n/2 ⌉−1 bits of advice are needed when the vertices on a path are presented for coloring in arbitrary order. The same holds in the more general case when just a subset of the vertices is colored online. However, the problem turns out to be non-trivial for the case where the online algorithm is guaranteed that the vertices it receives form a subset of a path and are presented in the order in which they lie on the path. For this variant we prove that its advice complexity is βn+O(logn) bits, where β≈0.406 is a fixed constant (we give its closed form). This suggests that the generalized problem will be challenging for more complex graph classes.

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