Abstract
This paper studies the kernelization complexity of graph coloring problems, with respect to certain structural parameterizations of the input instances. We are interested in how well polynomial-time data reduction can provably shrink instances of coloring problems, in terms of the chosen parameter. It is well known that deciding 3-colorability is already NP-complete, hence parameterizing by the requested number of colors is not fruitful. Instead, we pick up on a research thread initiated by Cai (DAM, 2003) who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial-time solvable; for example parameterizing by the number k of vertex-deletions needed to make the graph chordal. We obtain various upper and lower bounds for kernels of such parameterizations of q-Coloring, complementing Cai’s study of the time complexity with respect to these parameters.
This work was supported by the Netherlands Organization for Scientific Research (NWO), project “KERNELS: Combinatorial Analysis of Data Reduction”.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)
Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Cross-composition: A new technique for kernelization lower bounds. In: Proc. 28th STACS, pp. 165–176 (2011)
Bodlaender, H.L., Jansen, K., Woeginger, G.J.: Scheduling with incompatible jobs. Discrete Applied Mathematics 55(3), 219–232 (1994)
Bodlaender, H.L., Koster, A.M.C.A.: Combinatorial optimization on graphs of bounded treewidth. Comput. J. 51(3), 255–269 (2008)
Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. In: Proc. 17th ESA, pp. 635–646 (2009)
Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph classes: a survey. Society for Industrial and Applied Mathematics, Philadelphia (1999)
Cai, L.: Parameterized complexity of vertex colouring. Discrete Applied Mathematics 127(3), 415–429 (2003)
Chor, B., Fellows, M., Juedes, D.W.: Linear kernels in linear time, or how to save k colors in O(n 2) steps. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 257–269. Springer, Heidelberg (2004)
Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: Proc. 42nd STOC, pp. 251–260 (2010)
Downey, R., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999)
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. Inf. Comput. 209(2), 143–153 (2011)
Fellows, M.R., Lokshtanov, D., Misra, N., Mnich, M., Rosamond, F.A., Saurabh, S.: The complexity ecology of parameters: An illustration using bounded max leaf number. Theory Comput. Syst. 45(4), 822–848 (2009)
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)
Fomin, F.V., Golovach, P.A., Lokshtanov, D., Saurabh, S.: Intractability of clique-width parameterizations. SIAM J. Comput. 39(5), 1941–1956 (2010)
Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)
Jansen, B.M.P., Kratsch, S.: Data reduction for graph coloring problems. CoRR, abs/1104.4229 (2011)
Knuth, D.E.: Axioms and hulls. Springer, Heidelberg (1992)
Kobler, D., Rotics, U.: Edge dominating set and colorings on graphs with fixed clique-width. Discrete Applied Mathematics 126(2-3), 197–221 (2003)
Kratochvíl, J.: Precoloring extension with fixed color bound. Acta Mathematica Universitatis Comenianae 62(2), 139–153 (1993)
Lokshtanov, D., Marx, D., Saurabh, S.: Known algorithms on graphs of bounded treewidth are probably optimal. In: Proc. 22nd SODA, pp. 777–789 (2011)
Marx, D.: Parameterized coloring problems on chordal graphs. Theoretical Computer Science 351(3), 407–424 (2006)
Tuza, Z.: Graph colorings with local constraints - a survey. Math. Graph Theory 17, 161–228 (1997)
Yap, C.-K.: Some consequences of non-uniform conditions on uniform classes. Theor. Comput. Sci. 26, 287–300 (1983)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jansen, B.M.P., Kratsch, S. (2011). Data Reduction for Graph Coloring Problems. In: Owe, O., Steffen, M., Telle, J.A. (eds) Fundamentals of Computation Theory. FCT 2011. Lecture Notes in Computer Science, vol 6914. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22953-4_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-22953-4_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22952-7
Online ISBN: 978-3-642-22953-4
eBook Packages: Computer ScienceComputer Science (R0)