Parameterized Algorithms for <scp>Steiner Forest</scp> in Bounded Width Graphs

In this article, 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 et al. gave an approximation scheme with a run time of \(n^{O(k^{2}/{\varepsilon})}\) on graphs of treewidth \( k \) . Our main result is a much faster Efficient Parameterized Approximation Scheme (EPAS) with a run time of \(2^{O(\frac{k^{2}}{{\varepsilon}}\log\frac{k}{{\varepsilon}})}\,{\cdot}\,n^{O(1)}\) . 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 run-time 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.