Incremental List Coloring of Graphs, Parameterized by Conservation
Incrementally k-list coloring a graph means that a graph is given by adding stepwise one vertex after another, and for each intermediate step we ask for a vertex coloring such that each vertex has one of the colors specified by its associated list containing some of in total k colors. We introduce the “conservative version” of this problem by adding a further parameter c ∈ ℕ specifying the maximum number of vertices to be recolored between two subsequent graphs (differing by one vertex). This “conservation parameter” c models the natural quest for a modest evolution of the coloring in the course of the incremental process instead of performing radical changes. We show that the problem is NP-hard for k ≥ 3 and W-hard when parameterized by c. In contrast, the problem becomes fixed-parameter tractable with respect to the combined parameter (k,c). We prove that the problem has an exponential-size kernel with respect to (k,c) and there is no polynomial-size kernel unless NP ⊆ coNP/poly. Finally, we provide empirical findings for the practical relevance of our approach in terms of an effective graph coloring heuristic.
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- 1.Basu, S., Davidson, I., Wagstaff, K.: Constrained Clustering: Advances in Algorithms, Theory, and Applications. Chapman & Hall, Boca Raton (2008)Google Scholar
- 3.Bodlaender, H.L., Thomasse, S., Yeo, A.: Analysis of data reduction: Transformations give evidence for non-existence of polynomial kernels. Technical report, UU-CS-2008-030, Institute of Information and Computing Sciences, Utrecht University, Netherlands (2008)Google Scholar
- 5.Culberson, J.C., Luo, F.: Exploring the k-colorable landscape with iterated greedy. DIMACS Series in Discrete Math. and Theor. Comput. Sci., pp. 245–284 (1996)Google Scholar
- 6.DIMACS. Graph coloring and its generalizations (2002), http://mat.gsia.cmu.edu/COLOR02 (Accessed, December 2009)
- 7.DIMACS. Maximum clique, graph coloring, and satisfiability. Second DIMACS implementation challenge (1995), http://dimacs.rutgers.edu/Challenges/ (Accessed December 2009)
- 8.Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
- 9.Fellows, M.R., Fomin, F.V., Lokshtanov, D., Rosamond, F.A., Saurabh, S., Szeider, S., Thomassen, C.: On the complexity of some colorful problems parameterized by treewidth. In: Dress, A.W.M., Xu, Y., Zhu, B. (eds.) COCOA. LNCS, vol. 4616, pp. 366–377. Springer, Heidelberg (2007)CrossRefGoogle Scholar
- 11.Fellows, M.R., Rosamond, F.A., Fomin, F.V., Lokshtanov, D., Saurabh, S., Villanger, Y.: Local search: Is brute-force avoidable? In: Proc. 21st IJCAI, pp. 486–491 (2009b)Google Scholar
- 12.Fiala, J., Golovach, P.A., Kratochvíl, J.: Parameterized complexity of coloring problems: Treewidth versus vertex cover. In: Chen, J., Cooper, S.B. (eds.) TAMC 2009. LNCS, vol. 5532, pp. 221–230. Springer, Heidelberg (2009)Google Scholar
- 13.Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)Google Scholar
- 18.Shachnai, H., Tamir, G., Tamir, T.: Minimal cost reconfiguration of data placement in storage area network. In: Bampis, P. E. (ed.) WAOA 2009. LNCS, vol. 5893, pp. 229–241. Springer, Heidelberg (2010)Google Scholar