In the spanning-tree congestion problem ($\mathsf{STC}$), we are given a graph $G$, and the objective is to compute a spanning tree of $G$ that minimizes the maximum edge congestion. While $\mathsf{STC}$ is known to be $\mathbb{NP}$-hard, even for some restricted graph classes, several key questions regarding its computational complexity remain open, and we address some of these in our paper. (i) For graphs of degree at most $Δ$, it is known that $\mathsf{STC}$ is $\mathbb{NP}$-hard when $Δ\ge 8$. We provide a complete resolution of this variant, by showing that $\mathsf{STC}$ remains $\mathbb{NP}$-hard for each degree bound $Δ\ge 3$. (ii) In the decision version of $\mathsf{STC}$, given an integer $K$, the goal is to determine whether the congestion of $G$ is at most $K$. We prove that this variant is polynomial-time solvable for $K$-edge-connected graphs.

翻译：在生成树拥塞问题（$\mathsf{STC}$）中，给定一个图$G$，目标是计算$G$的一棵生成树，使得最大边拥塞最小。虽然已知$\mathsf{STC}$是$\mathbb{NP}$-难的，即使对于某些受限图类也是如此，但关于其计算复杂性的几个关键问题仍然悬而未决，我们在本文中解决了其中一些问题。(i) 对于最大度不超过$Δ$的图，已知当$Δ\ge 8$时$\mathsf{STC}$是$\mathbb{NP}$-难的。我们通过证明对于每个度数界$Δ\ge 3$，$\mathsf{STC}$仍然是$\mathbb{NP}$-难的，从而完全解决了该变体问题。(ii) 在$\mathsf{STC}$的判定版本中，给定一个整数$K$，目标是确定$G$的拥塞是否至多为$K$。我们证明对于$K$边连通图，该变体是多项式时间可解的。