Minimum entropy coloring

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.

[1]  Richard M. Karp,et al.  The minimum-entropy set cover problem , 2005, Theoretical Computer Science.

[2]  G. Pólya,et al.  Inequalities (Cambridge Mathematical Library) , 1934 .

[3]  Serge J. Belongie,et al.  On the non-optimality of four color coding of image partitions , 2002, Proceedings. International Conference on Image Processing.

[4]  Fabrizio Granelli,et al.  Efficient labeling procedures for image partition encoding , 2000, Signal Process..

[5]  H. S. WITSENHAUSEN,et al.  The zero-error side information problem and chromatic numbers (Corresp.) , 1976, IEEE Trans. Inf. Theory.

[6]  Uriel Feige,et al.  Zero Knowledge and the Chromatic Number , 1998, J. Comput. Syst. Sci..

[7]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[8]  Gary L. Miller,et al.  The Complexity of Coloring Circular Arcs and Chords , 1980, SIAM J. Algebraic Discret. Methods.

[9]  Paul Erdös,et al.  On the equality of the partial Grundy and upper ochromatic numbers of graphs , 2003, Discret. Math..

[10]  Rajiv Raman,et al.  Buffer minimization using max-coloring , 2004, SODA '04.

[11]  M. Golumbic.  Algorithmic graph theory and perfect graphs , 1980 .

[12]  Alan C. Tucker,et al.  An Efficient Test for Circular-Arc Graphs , 1980, SIAM J. Comput..

[13]  Jeff Kahn,et al.  Entropy and sorting , 1992, STOC '92.

[14]  Tommy R. Jensen,et al.  Graph Coloring Problems , 1994 .

[15]  Klaus Jansen,et al.  Approximation Results for the Optimum Cost Chromatic Partition Problem , 1997, J. Algorithms.

[16]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[17]  Michelle Effros,et al.  Low complexity code design for lossless and near-lossless side information source codes , 2003, Data Compression Conference, 2003. Proceedings. DCC 2003.

[18]  Jean Cardinal,et al.  On minimum entropy graph colorings , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

[19]  Dániel Marx.  A short proof of the NP-completeness of minimum sum interval coloring , 2005, Oper. Res. Lett..

[20]  V. Chvátal.  On certain polytopes associated with graphs , 1975 .

[21]  Alain Hertz,et al.  Feasible edge colorings of trees with cardinality constraints , 2000, Discret. Math..

[22]  D. de Werra,et al.  A chromatic scheduling model with costs , 1995 .

[23]  Noga Alon,et al.  Source coding and graph entropies , 1996, IEEE Trans. Inf. Theory.

[24]  R. L. Brooks.  On Colouring the Nodes of a Network , 1941 .

[25]  Harold N. Gabow,et al.  Data structures for weighted matching and nearest common ancestors with linking , 1990, SODA '90.

[26]  David Zuckerman.  Linear Degree Extractors and the Inapproximability of Max Clique and Chromatic Number , 2007, Theory Comput..