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：本文重新评估了在多种有界宽度图类中已被深入研究的Steiner森林问题的参数化复杂度与近似性。该问题以边加权图及顶点对作为输入，目标是找到一个最小代价子图，使得每个给定顶点对位于同一连通分量中。已知该问题在一般情况下是APX难的，在树宽为3、树深为4、反馈顶点集大小为2的图上均是NP难的。然而，Bateni、Hajiaghayi和Marx [JACM, 2011] 在树宽为$k$的图上给出了一个运行时间为$n^{O(\frac{k^2}{\varepsilon})}$的近似方案。我们的主要成果是一个更高效的参数化近似方案（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$k$的依赖是渐进最优的。此外，若$k$是反馈边集的大小，我们获得了更快的$2^{O(k)} \cdot n^{O(1)}$时间算法，该结果在ETH下同样不可改进。