Steiner Tree Parameterized by Multiway Cut and Even Less

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.