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.

This research was partially funded by the Networks programme via the Dutch Ministry of Education, Culture and Science through the Netherlands Organisation for Scientific Research. The research was done while the first author was at CWI, Amsterdam. The second author was supported by NWO Veni grant “Frontiers in Parameterized Preprocessing”.