On Structural Parameterizations of Happy Coloring, Empire Coloring and Boxicity
Distance parameters are extensively used to design efficient algorithms for many hard graph problems. They measure how far a graph is from belonging to some class of graphs. If a problem is tractable on a class of graphs Open image in new window , then distances to Open image in new window provide interesting parameterizations to that problem. For example, the parameter vertex cover measures the closeness of a graph to an edgeless graph. Many hard problems are tractable on graphs of small vertex cover. However, the parameter vertex cover is very restrictive in the sense that the class of graphs with bounded vertex cover is small. This significantly limits its usefulness in practical applications. In general, it is desirable to find tractable results for parameters such that the class of graphs with the parameter bounded should be as large as possible. In this spirit, we consider the parameter distance to threshold graphs, which are graphs that are both split graphs and cographs. It is a natural choice of an intermediate parameter between vertex cover and clique-width. In this paper, we give parameterized algorithms for some hard graph problems parameterized by the distance to threshold graphs. We show that Happy Coloring and Empire Coloring problems are fixed-parameter tractable when parameterized by the distance to threshold graphs. We also present an approximation algorithm to compute the Boxicity of a graph parameterized by the distance to threshold graphs.
KeywordsParameterized complexity Threshold graphs Algorithm
The authors are grateful to the anonymous referees for their valuable remarks and suggestions that significantly helped them improve the quality of the paper. The first author acknowledges support from Tata Consultancy Services (TCS) Research Fellowship.
- 2.Agrawal, A.: On the parameterized complexity of happy vertex coloring. In: International Workshop on Combinatorial Algorithms. Springer (2017, in press)Google Scholar
- 3.Aravind, N.R., Kalyanasundaram, S., Kare, A.S.: Linear time algorithms for happy vertex coloring problems for trees. In: Mäkinen, V., Puglisi, S.J., Salmela, L. (eds.) IWOCA 2016. LNCS, vol. 9843, pp. 281–292. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-44543-4_22 CrossRefGoogle Scholar
- 4.Aravind, N., Kalyanasundaram, S., Kare, A.S., Lauri, J.: Algorithms and hardness results for happy coloring problems. arXiv preprint arXiv:1705.08282 (2017)
- 5.Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM (1999)Google Scholar
- 7.Bulian, J.: Parameterized complexity of distances to sparse graph classes. Technical report, University of Cambridge, Computer Laboratory (2017)Google Scholar
- 10.Cozzens, M.B.: Higher and multi-dimensional analogues of interval graphs (1982)Google Scholar
- 18.Lampis, M.: Structural parameterizations of hard graph problems. City University of New York (2011)Google Scholar
- 21.Misra, N., Reddy, I.V.: The parameterized complexity of happy colorings. arXiv preprint arXiv:1708.03853 (2017)
- 22.Roberts, S.: On the boxicity and cubicity of a graph. In: Recent Progresses in Combinatorics, pp. 301–310 (1969)Google Scholar