In this paper we reassess the parameterized complexity and approximability of the well-studied Steiner Forest problem in several graph classes of bounded width. The problem takes an edge-weighted graph and pairs of vertices as input, and the aim is to find a minimum cost subgraph in which each given vertex pair lies in the same connected component. It is known that this problem is APX-hard in general, and NP-hard on graphs of treewidth 3, treedepth 4, and feedback vertex set size 2. However, Bateni, Hajiaghayi and Marx [JACM, 2011] gave an approximation scheme with a runtime of $n^{O(\frac{k^2}{\varepsilon})}$ on graphs of treewidth $k$. Our main result is a much faster efficient parameterized approximation scheme (EPAS) with a runtime of $2^{O(\frac{k^2}{\varepsilon} \log \frac{k^2}{\varepsilon})} \cdot n^{O(1)}$. If $k$ instead is the vertex cover number of the input graph, we show how to compute the optimum solution in $2^{O(k \log k)} \cdot n^{O(1)}$ time, and we also prove that this runtime dependence on $k$ is asymptotically best possible, under ETH. Furthermore, if $k$ is the size of a feedback edge set, then we obtain a faster $2^{O(k)} \cdot n^{O(1)}$ time algorithm, which again cannot be improved under ETH.

翻译：本文重新评估了多个有界宽度图类中经典Steiner森林问题的参数化复杂性和可近似性。该问题的输入是一个带权图及若干顶点对，目标是最小成本子图，使得每个给定顶点对均位于同一连通分量中。已知该问题在一般情况下为APX-难，且在树宽为3、树深为4及反馈顶点集大小为2的图上为NP-难。然而，Bateni、Hajiaghayi和Marx [JACM, 2011] 在树宽为$k$的图上给出了运行时间为$n^{O(\frac{k^2}{\varepsilon})}$的近似方案。我们的主要结果是更快速的参数化近似方案（EPAS），其运行时间为$2^{O(\frac{k^2}{\varepsilon} \log \frac{k^2}{\varepsilon})} \cdot n^{O(1)}$。若$k$为输入图的顶点覆盖数，我们展示了如何在$2^{O(k \log k)} \cdot n^{O(1)}$时间内计算最优解，并证明在ETH假设下该运行时间对$k$的依赖渐近最优。此外，当$k$为反馈边集大小时，我们得到更快的$2^{O(k)} \cdot n^{O(1)}$时间算法，该算法在ETH下同样无法改进。