Fine-Grained Parameterized Complexity Analysis of Graph Coloring Problems

The q-Coloring problem asks whether the vertices of a graph can be properly colored with q colors. Lokshtanov et al. [SODA 2011] showed that q-Coloring on graphs with a feedback vertex set of size k cannot be solved in time O∗((q− ε)), for any ε > 0, unless the Strong Exponential-Time Hypothesis (SETH) fails. In this paper we perform a fine-grained analysis of the complexity of q-Coloring with respect to a hierarchy of parameters. We show that even when parameterized by the vertex cover number, q must appear in the base of the exponent: Unless ETH fails, there is no universal constant θ such that q-Coloring parameterized by vertex cover can be solved in time O∗(θk) for all fixed q. We apply a method due to Jansen and Kratsch [Inform. & Comput. 2013] to prove that there are O∗((q− ε)) time algorithms where k is the vertex deletion distance to several graph classes F for which q-Coloring is known to be solvable in polynomial time. We generalize earlier ad-hoc results by showing that if F is a class of graphs whose (q + 1)-colorable members have bounded treedepth, then there exists some ε > 0 such that q-Coloring can be solved in time O∗((q−ε)k) when parameterized by the size of a given modulator to F . In contrast, we prove that if F is the class of paths – some of the simplest graphs of unbounded treedepth – then no such algorithm can exist unless SETH fails.