The {\em Spanning Tree Congestion} problem is an easy-to-state NP-hard problem: given a graph $G$, construct a spanning tree $T$ of $G$ minimizing its maximum edge congestion where the congestion of an edge $e\in T$ is the number of edges $uv$ in $G$ such that the unique path between $u$ and $v$ in $T$ passes through $e$; the optimum value for a given graph $G$ is denoted $STC(G)$. It is known that {\em every} spanning tree is an $n/2$-approximation. A long-standing problem is to design a better approximation algorithm. Our contribution towards this goal is an $O(\Delta\cdot\log^{3/2}n)$-approximation algorithm for the minimum congestion spanning tree problem where $\Delta$ is the maximum degree in $G$. For graphs with maximum degree bounded by polylog of the number of vertices, this is an exponential improvement over the previous best approximation. For graphs with maximum degree bounded by $o(n/\log^{3/2}n)$, we get $o(n)$-approximation; this is the largest class of graphs that we know of, for which sublinear approximation is known for this problem. Our main tool for the algorithm is a new lower bound on the spanning tree congestion which is of independent interest. We prove that for every graph $G$, $STC(G)\geq \Omega(hb(G)/\Delta)$ where $hb(G)$ denotes the maximum bisection width over all subgraphs of $G$.

翻译：{\em 生成树拥塞}问题是一个易于表述的NP难问题：给定图$G$，构造$G$的生成树$T$以最小化其最大边拥塞，其中边$e\in T$的拥塞定义为$G$中满足$u$和$v$在$T$中的唯一路径经过$e$的边$uv$的数量；对于给定图$G$的最优值记为$STC(G)$。已知{\em 任意}生成树都是$n/2$近似解。长期存在的问题是设计更好的近似算法。我们对此目标的贡献是提出了最小拥塞生成树问题的$O(\Delta\cdot\log^{3/2}n)$近似算法，其中$\Delta$是$G$中的最大度数。对于最大度数以顶点数的多对数有界的图，这相对于先前最佳近似是指数级改进。对于最大度数以$o(n/\log^{3/2}n)$有界的图，我们得到$o(n)$近似；这是目前已知对该问题具有亚线性近似的最广泛图类。我们算法的主要工具是一个新的生成树拥塞下界，其本身具有独立意义。我们证明对于任意图$G$，$STC(G)\geq \Omega(hb(G)/\Delta)$，其中$hb(G)$表示$G$的所有子图的最大二分宽度。