In this paper we study the Spanning Tree Congestion problem, where we are given a graph $G=(V,E)$ and are asked to find a spanning tree $T$ of minimum maximum congestion. Here, the congestion of an edge $e\in T$ is the number of edges $uv\in E$ such that the (unique) path from $u$ to $v$ in $T$ traverses $e$. We consider this well-studied NP-hard problem from the point of view of (structural) parameterized complexity and obtain the following results. We resolve a natural open problem by showing that Spanning Tree Congestion is not FPT parameterized by treewidth (under standard assumptions). More strongly, we present a generic reduction which applies to (almost) any parameter of the form ``vertex-deletion distance to class $\mathcal{C}$'', thus obtaining W[1]-hardness for parameters more restricted than treewidth, including tree-depth plus feedback vertex set, or incomparable to treewidth, such as twin cover. Via a slight tweak of the same reduction we also show that the problem is NP-complete on graphs of modular-width $4$. Even though it is known that Spanning Tree Congestion remains NP-hard on instances with only one vertex of unbounded degree, it is currently open whether the problem remains hard on bounded-degree graphs. We resolve this question by showing NP-hardness on graphs of maximum degree 8. Complementing the problem's W[1]-hardness for treewidth...

翻译：本文研究生成树拥塞问题：给定图$G=(V,E)$，需寻找最大拥塞值最小的生成树$T$。其中边$e\in T$的拥塞定义为满足$uv\in E$且$T$中$u$到$v$的唯一路径经过$e$的边数。我们从（结构）参数化复杂度视角研究这一经典NP难问题，获得以下结果：通过证明生成树拥塞在树宽参数化下不属于FPT（基于标准假设），解决了一个自然开放问题；进一步提出通用归约方法，适用于几乎所有“到类$\mathcal{C}$的顶点删除距离”形式参数，从而对严于树宽的参数（如树深加反馈顶点集）及与树宽不可比参数（如双覆盖）证明W[1]困难性；通过微调该归约，证明模宽为$4$的图上的NP完全性。尽管已知生成树拥塞在仅含一个无界度顶点的实例上仍为NP难，其在有界度图上的复杂性尚属开放。我们通过证明最大度$8$图上的NP难性解决了该问题。作为树宽参数W[1]困难性的补充...