Steiner Tree Parameterized by Multiway Cut and Even Less

TL;DR

This work studies Steiner Tree under two refined, terminal-aware FPT parameterizations. It provides a polynomial-space FPT algorithm parameterized by the size of a given multiway cut $S$, running in $2^{O(|S|\log|S|)}\cdot \mathrm{poly}(n)$ time, by reducing to minimum-cost bipartite matchings via $S$-connecting systems. It also introduces $K$-free treewidth, a hybrid parameter that refines both the terminal count and graph treewidth, and proves a single-exponential algorithm running in $2^{O(k)}\cdot \mathrm{poly}(n)$ with $k=\mathsf{tw}_{{K}}(G,K)$, leveraging $\mathcal{H}$-treewidth decompositions and rank-based DP. Together, these results broaden the parameterized toolkit for Steiner Tree by exploiting terminal-structure and provide efficient algorithms on graph classes where terminals interact with small separators.

Abstract

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.

Figures (2)

• Figure 1: The multiway cut $S= \{s_1,s_2,s_3,s_4,s_5,s_6,s_7\}$ ensures that no two terminals (squares) belong to the same connected component of $G-S$. The gray areas indicate the set $S$ and the components of $G-S$. The lines are a visual representation of a Steiner tree $F$ for the terminals, split at $S$, such that $F$ is partitioned into three trees of category 1 (solid trees, note the solid tree between $s_5$ and $s_6$), one tree of category 2 (dotted trees) and five paths of category 3 (dashed paths).
• Figure 2: In this example we have $X= \{x_1,\dots,x_8\}$. The lines are a visual representation of a subgraph $F$ of $G$, the graph $G$ itself is not drawn. For this subgraph $F$ we have $\pi_F(X) = \{\{x_1\},\{x_2, x_3, x_4, x_5\},\{x_6,x_7\},\{x_8\}\}$.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Definition 3: $S$-connecting system
• Definition 4: Self-reachable
• Lemma 4
• Lemma 5
• Lemma 6
• Theorem 6
• Definition 9: Tree $K$-free-decomposition
• Definition 10: Tree $\triangle$-free decomposition, Definition 5 of JansenKW23