Finer Tight Bounds for Coloring on Clique-Width
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We revisit the complexity of the classical k-Coloring problem parameterized by clique-width. This is a very well-studied problem that becomes highly intractable when the number of colors k is large. However, much less is known on its complexity for small, concrete values of k. In this paper, we completely determine the complexity of k-Coloring parameterized by clique-width for any fixed k, under the SETH. Specifically, we show that for all k> 3,ϵ>0, k-Coloring cannot be solved in time O^*((2^k-2-ϵ)^cw), and give an algorithm running in time O^*((2^k-2)^cw). Thus, if the SETH is true, 2^k-2 is the "correct" base of the exponent for every k. Along the way, we also consider the complexity of k-Coloring parameterized by the related parameter modular treewidth (mtw). In this case we show that the "correct" running time, under the SETH, is O^*(k k/2^mtw). If we base our results on a weaker assumption (the ETH), they imply that k-Coloring cannot be solved in time n^o(cw), even on instances with O( n) colors.
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