Courcelle's Theorem in Truly Linear FPT

Recently, Bumpus, Downey, Eagling-Vose, Enright, Fellows, Kutner, Larios-Jones, Martin, Rosamond, and Yates defined Truly Linear FPT (TLFPT) to be the class of parameterized problems with algorithms running in time $O(n) + f(k)$, where $n$ is the input size and $k$ the parameter [arXiv:2606.02492]. They gave several algorithmic techniques for designing TLFPT algorithms, but left parameterization by treewidth open. In this paper, we give a general method for designing TLFPT algorithms parameterized by treewidth, solving three open problems posed by Bumpus et al. In particular, we give a TLFPT algorithm for Courcelle's theorem: We show that given an $n$-vertex $m$-edge graph $G$, an integer $k$, and a $\mathsf{CMSO}_2$-formula $\varphi$, we can in time $O(n+m) + f(k, \varphi)$ either conclude that the treewidth of $G$ is more than $k$, or check whether $G$ satisfies $\varphi$. As a part of our algorithm, we give an approximation algorithm for treewidth that runs in time $O(n+m)$ and returns a tree decomposition whose width is at most $2^{O(k)}$ times the optimum. Our result also implies a TLFPT algorithm for computing the value of treewidth exactly.