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 (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.
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References
Accame M, De Natale FGB, Granelli F (2000) Efficient labeling procedures for image partition encoding. Signal Process 80(6):1127–1131
Agarwal S, Belongie S (2002) On the non-optimality of four color coding of image partitions. In: Proc. IEEE int. conf. image processing
Alon N, Orlitsky A (1996) Source coding and graph entropies. IEEE Trans Inf Theory 42(5):1329–1339
Brooks RL (1941) On colouring the nodes of a network. Proc Camb Philos Soc 37:194–197
Cardinal J, Fiorini S, Van Assche G (2004) On minimum entropy graph colorings. In: Proc. IEEE int. symposium on information theory, p 43
Chvátal V (1975) On certain polytopes associated with graphs. J Comb Theory Ser B 18:138–154
de Werra D, Glover F, Silver EA (1995) A chromatic scheduling model with costs. IIE Trans 27:181–189
de Werra D, Hertz A, Kobler D, Mahadev NVR (2000) Feasible edge colorings of trees with cardinality constraints. Discrete Math 222(1-3):61–72
Diestel R (2000) Graph theory, 2nd edn. Graduate Texts in Mathematics, vol 173. Springer, New York
Erdös P, Hedetniemi ST, Laskar RC, Prins GCE (2003) On the equality of the partial Grundy and upper ochromatic numbers of graphs. Discrete Math 272(1):53–64. In honor of Frank Harary
Feige U, Kilian J (1998) Zero knowledge and the chromatic number. J Comput Syst Sci 57(2):187–199
Gabow HN (1990) Data structures for weighted matching and nearest common ancestors with linking. In: 1st annual ACM-SIAM symposium on discrete algorithms, pp 434–443
Garey MR, Johnson DS, Miller GL, Papadimitriou CH (1980) The complexity of coloring circular arcs and chords. SIAM J Algebr Discrete Methods 1(2):216–227
Golumbic MC (2004) Algorithmic graph theory and perfect graphs. Academic Press, New York
Grötschel M, Lovász L, Schrijver A (1993) Geometric algorithms and combinatorial optimization, 2nd edn. Algorithms and Combinatorics, vol 2. Springer, Berlin
Halperin E, Karp RM (2004) The minimum-entropy set cover problem. In: Automata, languages and programming. Lecture notes in comput. sci., vol 3142. Springer, Berlin, pp 733–744
Hardy GH, Littlewood JE, Pólya G (1988) Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge. Reprint of the 1952 edition
Jansen K (2000) Approximation results for the optimum cost chromatic partition problem. J Algorithms 34:54–89
Jensen TR, Toft B (1995) Graph coloring problems. Wiley–Interscience, New York
Kahn J, Kim JH (1995) Entropy and sorting. J Comput Syst Sci 51(3):390–399
Körner J (1973) Coding of an information source having ambiguous alphabet and the entropy of graphs. In: Transactions of the sixth Prague conference on information theory, statistical decision functions, random processes (Tech Univ., Prague, 1971; dedicated to the memory of Antonín Špaček). Academia, Prague, pp 411–425
Marx D (2005) A short proof of the NP-completeness of minimum sum interval coloring. Oper Res Lett 33(4):382–384
Pemmaraju SV, Raman R, Varadarajan K (2004) Buffer minimization using max-coloring. In: SODA ’04: Proceedings of the fifteenth annual ACM-SIAM symposium on discrete algorithms. Siam, Philadelphia, pp 562–571
Simonyi G (2001) Perfect graphs and graph entropy. An updated survey. In: Perfect graphs. Discrete math. optim. Wiley, Chichester, pp 293–328
Tucker A (1980) An efficient test for circular-arc graphs. SIAM J Comput 9(1):1–24
Witsenhausen HS (1976) The zero-error side information problem and chromatic numbers. IEEE Trans Inf Theory 22(5):592–593
Zhao Q, Effros M (2003) Low complexity code design for lossless and near-lossless side information source codes. In: Proc. IEEE data compression conf.
Zuckerman D (2007) Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput 3(6):103–128
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S. Fiorini acknowledges the support from the Fonds National de la Recherche Scientifique and GERAD-HEC Montréal.
G. Joret is a F.R.S.-FNRS Research Fellow.
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Cardinal, J., Fiorini, S. & Joret, G. Minimum entropy coloring. J Comb Optim 16, 361–377 (2008). https://doi.org/10.1007/s10878-008-9152-2
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DOI: https://doi.org/10.1007/s10878-008-9152-2