Finer Tight Bounds for Coloring on Clique-Width

TL;DR

This paper precisely characterizes the computational complexity of the k-Coloring problem parameterized by clique-width for fixed k, establishing tight bounds under the SETH and exploring related parameters like modular treewidth.

Contribution

It provides exact exponential time bounds for k-Coloring based on clique-width and modular treewidth, confirming the optimality of these bounds under SETH.

Findings

For all k ≥ 3, k-Coloring cannot be solved faster than (2^k - 2 - ε)^{cw} under SETH.

An algorithm with running time (2^k - 2)^{cw} is presented, matching the lower bound.

The results extend to modular treewidth, with tight bounds based on binomial coefficients.

Abstract

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 $k\ge 3,\epsilon>0$, $k$-Coloring cannot be solved in time $O^*((2^k-2-\epsilon)^{cw})$, 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 $O^*({k\choose \lfloor k/2\rfloor}^{mtw})$. 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 $O(\log n)$ colors.