In length-constrained minimum spanning tree (MST) we are given an $n$-node graph $G = (V,E)$ with edge weights $w : E \to \mathbb{Z}_{\geq 0}$ and edge lengths $l: E \to \mathbb{Z}_{\geq 0}$ along with a root node $r \in V$ and a length-constraint $h \in \mathbb{Z}_{\geq 0}$. Our goal is to output a spanning tree of minimum weight according to $w$ in which every node is at distance at most $h$ from $r$ according to $l$. We give a polynomial-time algorithm for planar graphs which, for any constant $ε> 0$, outputs an $O\left(\log^{1+ε} n\right)$-approximate solution with every node at distance at most $(1+ε)h$ from $r$ for any constant $ε> 0$. Our algorithm is based on new length-constrained versions of classic planar separators which may be of independent interest. Additionally, our algorithm works for length-constrained Steiner tree. Complementing this, we show that any algorithm on general graphs for length-constrained MST in which nodes are at most $2h$ from $r$ cannot achieve an approximation of $O\left(\log ^{2-ε} n\right)$ for any constant $ε> 0$ under standard complexity assumptions; as such, our results separate the approximability of length-constrained MST in planar and general graphs.

翻译：在长度约束最小生成树问题中，给定一个具有$n$个节点的图$G = (V,E)$，其边权重函数为$w : E \to \mathbb{Z}_{\geq 0}$，边长度函数为$l: E \to \mathbb{Z}_{\geq 0}$，同时给定根节点$r \in V$和长度约束$h \in \mathbb{Z}_{\geq 0}$。我们的目标是输出一棵根据$w$计算权重最小的生成树，且其中每个节点根据$l$计算到$r$的距离不超过$h$。针对平面图，我们提出了一种多项式时间算法，对于任意常数$ε> 0$，该算法能够输出一个$O\left(\log^{1+ε} n\right)$近似解，且其中每个节点到$r$的距离不超过$(1+ε)h$。我们的算法基于经典平面分离器的新型长度约束变体，这些变体可能具有独立的研究价值。此外，该算法同样适用于长度约束Steiner树问题。作为补充，我们证明在标准复杂性假设下，针对一般图的长度约束最小生成树问题，任何保证节点到$r$距离不超过$2h$的算法均无法实现$O\left(\log ^{2-ε} n\right)$的近似比（其中$ε> 0$为任意常数）；因此，我们的结果从可近似性角度区分了平面图与一般图中长度约束最小生成树问题的难度差异。