Parameterized algorithms for Steiner Forest 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 2 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 aruntimeof ongraphsoftreewidth .OurmainresultisamuchfasterEfficientParameterizedApproximation Scheme (EPAS) with a run time of 2 2 log 1 . If instead is the vertex cover number of the input graph, we show how to compute the optimum solution in 2 log 1 time, and we also prove that this run-time dependence on isasymptotically best possible, under ETH. Furthermore, if is the size of a feedback edge set, then we obtain a faster 2 1 timealgorithm, which again cannot be improved under ETH.