The Spanning Tree Congestion (STC) problem is the following 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 optimal value for a given graph $G$ is denoted $STC(G)$. It is known that every spanning tree is an $n/2$-approximation for the STP problem. A long-standing problem is to design a better approximation algorithm. Our contribution towards this goal is an $O(Δ\cdot\log^{3/2}n)$-approximation algorithm where $Δ$ is the maximum degree in $G$ and $n$ the number of vertices. For graphs with a maximum degree bounded by a polylog of the number of vertices, this is an exponential improvement over the previous best approximation. Our main tool for the algorithm is a new lower bound on the spanning tree congestion which is of independent interest. Denoting by $hb(G)$ the hereditary bisection of $G$ which is the maximum bisection width over all subgraphs of $G$, we prove that for every graph $G$, $STC(G)\geq Ω(hb(G)/Δ)$.

翻译：生成树拥塞（STC）问题是以下NP难问题：给定图$G$，构造$G$的一棵生成树$T$，最小化其最大边拥塞，其中边$e\in T$的拥塞是指$G$中满足$u$与$v$在$T$中唯一路径经过$e$的边$uv$的数量；给定图$G$的最优值记为$STC(G)$。已知每一棵生成树都是STP问题的$n/2$近似。一个长期悬而未决的问题是设计更好的近似算法。我们对此的贡献是一个$O(Δ\cdot\log^{3/2}n)$近似算法，其中$Δ$是$G$的最大度数，$n$为顶点数。对于最大度数受顶点数多项式对数界定的图，这相比先前最佳近似实现了指数级改进。我们算法的核心工具是一个关于生成树拥塞的新下界，该下界具有独立意义。用$hb(G)$表示$G$的遗传二分法，即$G$所有子图上的最大二分宽度，我们证明对于每个图$G$，有$STC(G)\geq Ω(hb(G)/Δ)$。