Parameterized Algorithms for Steiner Forest in Bounded Width Graphs
Andreas Emil Feldmann, Michael Lampis
TL;DR
This work analyzes Steiner Forest on graphs of bounded width from a parameterized perspective, delivering an efficient parameterized approximation scheme (EPAS) for treewidth with runtime $2^{O( rac{k^2}{ rac{1}{ ext{}} ext{ }oldsymbol{ aggedleft rac{ ext{}}{}}})} $ (formatted correctly in the final document) and establishing ETH-tight FPT bounds for vertex cover and feedback-edge-set parameters: $2^{O(k\log k)}\cdot n^{O(1)}$ and $2^{O(k)}\cdot n^{O(1)}$, respectively. The approach extends Bateni et al.'s partition-based DP by incorporating a height parameter and $oldsymbol{\delta}$-net based refinement of active-terminal partitions, paired with Bodlaender's height-reduction to achieve FPT running times in both width and accuracy. The results close the gap between XP-approximation and FPT exactness for Steiner Forest on bounded-width graphs, and include ETH-based lower bounds that match the achieved dependencies, even for planarity variants and related width-parameterizations. The techniques yield practical implications for planar and minor-closed graph classes by enabling faster, near-optimal Steiner Forest computations via structured decompositions and targeted DP constructions.
Abstract
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.