Given a graph $G=(V,E)$ with a root $r\in V$, positive capacities $\left\{c\left(e\right)\right|e\in E\}$, and non-negative lengths $\left\{\ell \left(e\right)\right|e\in E\}$, the minimum-length (rooted) edge capacitated Steiner tree problem is to find a tree in $G$ of minimum total length, rooted at $r$, spanning a given subset $T\subset V$ of vertices, and such that, for each $e\in E$, there are at most $c\left(e\right)$ paths, linking $r$ to vertices in $T$, that contain $e$. We study the complexity and approximability of the problem, considering several relevant parameters such as the number of terminals, the edge lengths and the minimum and maximum edge capacities. For all but one combinations of assumptions regarding these parameters, we settle the question, giving a complete characterization that separates tractable cases from hard ones. The only remaining open case is proved to be equivalent to a long-standing open problem. We also prove close relations between our problem and classic Steiner tree as well as vertex-disjoint paths problems.

给定图 $G=（V，Ë）$ 有根 $\mathrm{[R}\in V$，积极的能力 $\left\{C（Ë）\right|Ë\in Ë\}$，以及非负长度 $\left\{\ell （Ë）\right|Ë\in Ë\}$，最小长度（有根）边缘容限的Steiner树问题是在 $G$ 最小总长度，植根于 $\mathrm{[R}$，跨越给定的子集 $Ť\subset V$ 的顶点，对于每个顶点 $Ë\in Ë$，最多 $C（Ë）$ 路径，链接 $\mathrm{[R}$ 在顶点 $Ť$，其中包含 $Ë$。我们考虑了几个相关参数，例如端子数量，边缘长度以及最小和最大边缘容量，研究了问题的复杂性和可近似性。除了关于这些参数的假设的所有组合以外，我们解决了这个问题，给出了将易处理案例与难解决案例区分开的完整特征。唯一剩下的未解决问题被证明等同于一个长期存在的未解决问题。我们还证明了我们的问题与经典Steiner树以及不相交的路径问题之间的密切关系。