In the spanning tree congestion problem, given a connected graph $G$, the objective is to compute a spanning tree $T$ in $G$ that minimizes its maximum edge congestion, where the congestion of an edge $e$ of $T$ is the number of edges in $G$ for which the unique path in $T$ between their endpoints traverses $e$. The problem is known to be $\mathbb{NP}$-hard, but its approximability is still poorly understood. In the decision version of this problem, denoted $K-\textsf{STC}$, we need to determine if $G$ has a spanning tree with congestion at most $K$. It is known that $K-\textsf{STC}$ is $\mathbb{NP}$-complete for $K\ge 8$. On the other hand, $3-\textsf{STC}$ can be solved in polynomial time, with the complexity status of this problem for $K\in \{4,5,6,7\}$ remaining an open problem. We substantially improve the earlier hardness results by proving that $K-\textsf{STC}$ is $\mathbb{NP}$-complete for $K\ge 5$. This leaves only the case $K=4$ open, and improves the lower bound on the approximation ratio to $1.2$. Motivated by evidence that minimizing congestion is hard even for graphs of small constant radius, we consider $K-\textsf{STC}$ restricted to graphs of radius $2$, and we prove that this variant is $\mathbb{NP}$-complete for all $K\ge 6$. Exploring further in this direction, we also examine the variant, denoted $K-\textsf{STC}D$, where the objective is to determine if the graph has a depth-$D$ spanning three of congestion at most $K$. We prove that $6-\textsf{STC}2$ is $\mathbb{NP}$-complete even for bipartite graphs. For bipartite graphs we establish a tight bound, by also proving that $5-\textsf{STC}2$ is polynomial-time solvable. Additionally, we complement this result with polynomial-time algorithms for two special cases that involve bipartite graphs and restrictions on vertex degrees.

翻译：在生成树拥塞问题中，给定连通图$G$，目标是计算$G$中的一棵生成树$T$，使其最大边拥塞度最小化，其中$T$的边$e$的拥塞度定义为$G$中端点间唯一路径经过$e$的边数。该问题已知为$\mathbb{NP}$-难，但其可逼近性仍知之甚少。在该问题的判定版本（记为$K-\textsf{STC}$）中，我们需要判断$G$是否存在拥塞度不超过$K$的生成树。已知当$K\ge 8$时，$K-\textsf{STC}$是$\mathbb{NP}$-完全的；另一方面，$3-\textsf{STC}$可在多项式时间内求解，而$K\in \{4,5,6,7\}$时的复杂度状态仍是开放问题。我们通过证明$K\ge 5$时$K-\textsf{STC}$是$\mathbb{NP}$-完全的，显著改进了先前的硬度结果。这仅留下$K=4$的情况未解，并将近似比下界提升至$1.2$。鉴于即使对于常数小半径的图，最小化拥塞度也难以处理的证据，我们考虑限制于半径$2$的图的$K-\textsf{STC}$，并证明该变体对所有$K\ge 6$是$\mathbb{NP}$-完全的。沿此方向进一步探索，我们还研究了记为$K-\textsf{STC}D$的变体，其目标是判断图是否存在深度为$D$且拥塞度不超过$K$的生成树。我们证明即使对于二分图，$6-\textsf{STC}2$也是$\mathbb{NP}$-完全的。对于二分图，我们通过同时证明$5-\textsf{STC}2$可在多项式时间内求解，建立了紧界。此外，我们针对涉及二分图和顶点度限制的两个特殊情形，以多项式时间算法补充了这一结果。