Fine-Grained Parameterized Complexity Analysis of Graph Coloring Problems

Abstract

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 \(\mathcal {O}^*((q-\varepsilon )^k)\), for any \(\varepsilon > 0\), unless the Strong Exponential-Time Hypothesis (\(\mathsf{SETH}\)) 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 unless \(\mathsf{ETH}\) fails, there is no universal constant \(\theta \) such that q-Coloring parameterized by vertex cover can be solved in time \(\mathcal {O}^*(\theta ^k)\) for all fixed q. We prove that there are \(\mathcal {O}^*((q - \varepsilon )^k)\) time algorithms where k is the vertex deletion distance to several graph classes \(\mathcal {F}\) for which q-Coloring is known to be solvable in polynomial time, including all graph classes whose \((q+1)\)-colorable members have bounded treedepth. In contrast, we prove that if \(\mathcal {F}\) is the class of paths – some of the simplest graphs of unbounded treedepth – then no such algorithm can exist unless \(\mathsf{SETH}\) fails.