We provide the first algorithm for computing an optimal tree decomposition for a given graph $G$ that runs in single exponential time in the feedback vertex number of $G$, that is, in time $2^{O(\text{fvn}(G))}\cdot n^{O(1)}$, where $\text{fvn}(G)$ is the feedback vertex number of $G$ and $n$ is the number of vertices of $G$. On a classification level, this improves the previously known results by Chapelle et al. [Discrete Applied Mathematics '17] and Fomin et al. [Algorithmica '18], who independently showed that an optimal tree decomposition can be computed in single exponential time in the vertex cover number of $G$. One of the biggest open problems in the area of parameterized complexity is whether we can compute an optimal tree decomposition in single exponential time in the treewidth of the input graph. The currently best known algorithm by Korhonen and Lokshtanov [STOC '23] runs in $2^{O(\text{tw}(G)^2)}\cdot n^4$ time, where $\text{tw}(G)$ is the treewidth of $G$. Our algorithm improves upon this result on graphs $G$ where $\text{fvn}(G)\in o(\text{tw}(G)^2)$. On a different note, since $\text{fvn}(G)$ is an upper bound on $\text{tw}(G)$, our algorithm can also be seen either as an important step towards a positive resolution of the above-mentioned open problem, or, if its answer is negative, then a mark of the tractability border of single exponential time algorithms for the computation of treewidth.

翻译：我们提出了首个在给定图$G$的反馈顶点数上以单指数时间运行的最优树分解计算算法，即该算法的时间复杂度为$2^{O(\text{fvn}(G))}\cdot n^{O(1)}$，其中$\text{fvn}(G)$表示$G$的反馈顶点数，$n$为$G$的顶点数。在分类层面，这一结果改进了此前Chapelle等人[Discrete Applied Mathematics '17]与Fomin等人[Algorithmica '18]独立证明的最优树分解可在$G$的顶点覆盖数上以单指数时间计算的结果。参数化复杂性领域最大的开放问题之一，在于能否在输入图树宽度的单指数时间内计算出最优树分解。当前Korhonen与Lokshtanov [STOC '23]提出的最佳已知算法运行时间为$2^{O(\text{tw}(G)^2)}\cdot n^4$，其中$\text{tw}(G)$为$G$的树宽度。在满足$\text{fvn}(G)\in o(\text{tw}(G)^2)$的图$G$上，我们的算法改进了这一结果。另一方面，鉴于$\text{fvn}(G)$是$\text{tw}(G)$的上界，本算法既可视为向上述开放问题的正向解决迈出的重要一步，亦可在其答案是否定的情况下，成为树宽度计算中单指数时间算法可解性边界的标志。