AI 中文总结
本文针对树宽参数化设计真线性固定参数可解（TLFPT）算法，解决了三个开放问题。给出库采尔定理的TLFPT算法，还给出运行时间\(O(n + m)\)的树宽近似算法，其宽度至多为最优值\(2^{O(k)}\)倍，也暗示了精确计算树宽值的TLFPT算法。
AI 中文摘要
最近，Bumpus等人定义真线性固定参数可解（TLFPT）为算法运行时间为\(O(n)+f(k)\)的参数化问题类，其中\(n\)是输入规模，\(k\)是参数。他们给出了设计TLFPT算法的几种技术，但树宽参数化问题未解决。本文给出了一种由树宽参数化设计TLFPT算法的通用方法，解决了Bumpus等人提出的三个开放问题。具体而言，给出了库采尔定理的TLFPT算法：给定一个\(n\)个顶点\(m\)条边的图\(G\)、整数\(k\)和一个\(\mathsf{CMSO}_2\)公式\(\varphi\)，能在时间\(O(n + m)+f(k,\varphi)\)内判定\(G\)的树宽是否大于\(k\)，或检查\(G\)是否满足\(\varphi\)。作为算法一部分，给出了一个树宽近似算法，运行时间为\(O(n + m)\)，返回宽度至多为最优值\(2^{O(k)}\)倍的树分解。结果还暗示了精确计算树宽值的TLFPT算法。
英文摘要
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 $φ$, we can in time $O(n+m) + f(k, φ)$ either conclude that the treewidth of $G$ is more than $k$, or check whether $G$ satisfies $φ$. 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.
Comments18 pages
