Journal of Combinatorial Optimization

, Volume 16, Issue 4, pp 361–377 | Cite as

Minimum entropy coloring

  • Jean Cardinal
  • Samuel Fiorini
  • Gwenaël Joret


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 (IEEE Trans. Inform. Theory 42(5):1329–1339, 1996) 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, for every constant ε>0, it is NP-hard to find a coloring whose entropy is within (1−ε)log n of the chromatic entropy, where n is the number of vertices of the graph. A simple polynomial case is also identified. It is known that 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.


Complexity Entropy Graph coloring 


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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Computer Science Dept., CP 212Université Libre de Bruxelles (U.L.B.)BrusselsBelgium
  2. 2.Mathematics Dept., CP 216Université Libre de Bruxelles (U.L.B.)BrusselsBelgium

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