Let c, k be two positive integers. Given a graph G=(V,E), the c-Load Coloring problem asks whether there is a c-coloring varphi: V => [c] such that for every i in [c], there are at least k edges with both endvertices colored i. Gutin and Jones (IPL 2014) studied this problem with c=2. They showed 2-Load Coloring to be fixed-parameter tractable (FPT) with parameter k by obtaining a kernel with at most 7k vertices. In this paper, we extend the study to any fixed c by giving both a linear-vertex and a linear-edge kernel. In the particular case of c=2, we obtain a kernel with less than 4k vertices and less than 8k edges. These results imply that for any fixed c >= 2, c-Load Coloring is FPT and the optimization version of c-Load Coloring (where k is to be maximized) has an approximation algorithm with a constant ratio.
@InProceedings{barbero_et_al:LIPIcs.IPEC.2015.43, author = {Barbero, Florian and Gutin, Gregory and Jones, Mark and Sheng, Bin}, title = {{Parameterized and Approximation Algorithms for the Load Coloring Problem}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {43--54}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.43}, URN = {urn:nbn:de:0030-drops-55703}, doi = {10.4230/LIPIcs.IPEC.2015.43}, annote = {Keywords: Load Coloring, fixed-parameter tractability, kernelization} }
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