Abstract
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.
The research is partially funded by the SNF grant 200021-132510/1 and by VEGA grants 1/0726/09 and 1/0979/12.
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Forišek, M., Keller, L., Steinová, M. (2012). Advice Complexity of Online Coloring for Paths. In: Dediu, AH., Martín-Vide, C. (eds) Language and Automata Theory and Applications. LATA 2012. Lecture Notes in Computer Science, vol 7183. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28332-1_20
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DOI: https://doi.org/10.1007/978-3-642-28332-1_20
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