Abstract
We study an information-theoretic variant of the graph coloring problem in which the objective function to minimize is the entropy of the coloring. The minimum entropy of a coloring is called the chromatic entropy and was shown by Alon and Orlitsky to play a fundamental role in the problem of coding with side information. In this paper, we consider the minimum entropy coloring problem from a computational point of view. We first prove that this problem is NP-hard on interval graphs. We then show that it is NP-hard to find a coloring whose entropy is within \(({1 \over 7} - \epsilon){\rm log} n\) of the chromatic entropy for any ε > 0, where n is the number of vertices of the graph. A simple polynomial case is also identified. It is known that the traditional graph entropy is a lower bound for the chromatic entropy. We prove that this bound can be arbitrarily bad, even for chordal graphs. Finally, we consider the minimum number of colors required to achieve minimum entropy and prove a Brooks-type theorem.
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Cardinal, J., Fiorini, S., Joret, G. (2005). Minimum Entropy Coloring. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_82
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DOI: https://doi.org/10.1007/11602613_82
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30935-2
Online ISBN: 978-3-540-32426-3
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