In the Steiner Tree problem we are given an undirected edge-weighted graph as input, along with a set $K$ of vertices called terminals. The task is to output a minimum-weight connected subgraph that spans all the terminals. The famous Dreyfus-Wagner algorithm running in $3^{|K|} \mathsf{poly}(n)$ time shows that the problem is fixed-parameter tractable parameterized by the number of terminals. We present fixed-parameter tractable algorithms for Steiner Tree using structurally smaller parameterizations. Our first result concerns the parameterization by a multiway cut $S$ of the terminals, which is a vertex set $S$ (possibly containing terminals) such that each connected component of $G-S$ contains at most one terminal. We show that Steiner Tree can be solved in $2^{O(|S|\log|S|)}\mathsf{poly}(n)$ time and polynomial space, where $S$ is a minimum multiway cut for $K$. The algorithm is based on the insight that, after guessing how an optimal Steiner tree interacts with a multiway cut $S$, computing a minimum-cost solution of this type can be formulated as minimum-cost bipartite matching. Our second result concerns a new hybrid parameterization called $K$-free treewidth that simultaneously refines the number of terminals $|K|$ and the treewidth of the input graph. By utilizing recent work on $\mathcal{H}$-Treewidth in order to find a corresponding decomposition of the graph, we give an algorithm that solves Steiner Tree in time $2^{O(k)} \mathsf{poly}(n)$, where $k$ denotes the $K$-free treewidth of the input graph. To obtain this running time, we show how the rank-based approach for solving Steiner Tree parameterized by treewidth can be extended to work in the setting of $K$-free treewidth, by exploiting existing algorithms parameterized by $|K|$ to compute the table entries of leaf bags of a tree $K$-free decomposition.

翻译：在斯坦纳树问题中，我们以无向带权边图作为输入，同时给定一个称为终端的顶点集合$K$。任务的目标是输出一个连接所有终端的最小权重连通子图。著名的Dreyfus-Wagner算法以$3^{|K|} \mathsf{poly}(n)$时间运行，表明该问题在终端数量参数化下是固定参数可解的。本文提出了使用结构更小的参数化的斯坦纳树固定参数可解算法。我们的第一个结果涉及以终端的多路割$S$作为参数，其中$S$是一个顶点集合（可能包含终端），使得$G-S$的每个连通分量至多包含一个终端。我们证明斯坦纳树可在$2^{O(|S|\log|S|)}\mathsf{poly}(n)$时间和多项式空间内求解，其中$S$是$K$的最小多路割。该算法的核心洞见在于：在推测最优斯坦纳树与多路割$S$的交互方式后，计算此类最小代价解可转化为最小代价二分图匹配问题。我们的第二个结果涉及一种称为$K$-自由树宽的新型混合参数化，该参数化同时优化了终端数量$|K|$和输入图的树宽。通过利用近期关于$\mathcal{H}$-树宽的研究来寻找图的对应分解，我们给出了一个在$2^{O(k)} \mathsf{poly}(n)$时间内求解斯坦纳树的算法，其中$k$表示输入图的$K$-自由树宽。为达到该时间复杂度，我们展示了如何将基于秩的树宽参数化斯坦纳树求解方法扩展至$K$-自由树宽场景：通过利用现有的$|K|$参数化算法来计算树$K$-自由分解中叶节点的表格项。