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 interval 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 $G=(V,E)$，要求寻找一棵生成树 $T$ 使得其最大拥塞最小化。其中，边 $e\in T$ 的拥塞定义为满足条件（在 $T$ 中从 $u$ 到 $v$ 的唯一路径经过 $e$）的边 $uv\in E$ 的数量。我们从（结构）参数化复杂度的角度研究这一已被充分研究的NP难问题，并得到以下结果。我们通过证明生成树拥塞问题在树宽参数下（在标准假设下）不是FPT，解决了一个自然的开放问题。更进一步，我们提出一个通用归约方法，适用于形如"到类 $\mathcal{C}$ 的顶点删除距离"的（几乎）所有参数，从而获得了对于比树宽更受限的参数（例如树深加上反馈顶点集）或与树宽不可比较的参数（例如双子覆盖）的W[1]难度。通过对同一归约进行略微调整，我们还证明了该问题在模宽度为4的区间图上是NP完全的。尽管已知生成树拥塞问题在仅有一个无界度顶点的实例上仍然是NP难的，但目前尚不清楚该问题在有界度图上是否仍然困难。我们通过证明最大度为8的图上的NP难度解决了这一问题。作为该问题在树宽参数下W[1]难度的补充...