In their seminal paper, Alth\"{o}fer et al. (DCG 1993) introduced the {\em greedy spanner} and showed that, for any weighted planar graph $G$, the weight of the greedy $(1+\epsilon)$-spanner is at most $(1+\frac{2}{\epsilon}) \cdot w(MST(G))$, where $w(MST(G))$ is the weight of a minimum spanning tree $MST(G)$ of $G$. This bound is optimal in an {\em existential sense}: there exist planar graphs $G$ for which any $(1+\epsilon)$-spanner has a weight of at least $(1+\frac{2}{\epsilon}) \cdot w(MST(G))$. However, as an {\em approximation algorithm}, even for a {\em bicriteria} approximation, the weight approximation factor of the greedy spanner is essentially as large as the existential bound: There exist planar graphs $G$ for which the greedy $(1+x \epsilon)$-spanner (for any $1\leq x = O(\epsilon^{-1/2})$) has a weight of $\Omega(\frac{1}{\epsilon \cdot x^2})\cdot w(G_{OPT, \epsilon})$, where $G_{OPT, \epsilon}$ is a $(1+\epsilon)$-spanner of $G$ of minimum weight. Despite the flurry of works over the past three decades on approximation algorithms for spanners as well as on light(-weight) spanners, there is still no (possibly bicriteria) approximation algorithm for light spanners in weighted planar graphs that outperforms the existential bound. As our main contribution, we present a polynomial time algorithm for constructing, in any weighted planar graph $G$, a $(1+\epsilon\cdot 2^{O(\log^* 1/\epsilon)})$-spanner for $G$ of total weight $O(1)\cdot w(G_{OPT, \epsilon})$. To achieve this result, we develop a new technique, which we refer to as {\em iterative planar pruning}. It iteratively modifies a spanner [...]

翻译：在Althöfer等人（DCG 1993）的开创性论文中，他们引入了贪心生成子的概念，并证明对于任意带权平面图G，贪心(1+ε)-生成子的权重至多为(1+2/ε)·w(MST(G))，其中w(MST(G))是G的最小生成树MST(G)的权重。该界限在存在性意义下是最优的：存在某些平面图G，其任意(1+ε)-生成子的权重至少为(1+2/ε)·w(MST(G))。然而，作为一种近似算法，即使是双准则近似，贪心生成子的权重近似因子本质上与存在性界限一样大：存在某些平面图G，使得贪心(1+xε)-生成子（对于任意满足1≤x=O(ε^{-1/2})的x）的权重达到Ω(1/(ε·x^2))·w(G_{OPT, ε})，其中G_{OPT, ε}是G的最小权重的(1+ε)-生成子。尽管过去三十年间在生成子的近似算法以及轻量生成子方面涌现了大量研究，但对于带权平面图中的轻量生成子，目前仍没有（可能是双准则的）近似算法能够超越存在性界限。作为我们的主要贡献，我们提出了一种多项式时间算法，该算法能在任意带权平面图G中构造一个(1+ε·2^{O(log^* 1/ε)})-生成子，其总权重为O(1)·w(G_{OPT, ε})。为实现这一结果，我们发展了一种新技术，称之为迭代平面剪枝。该技术通过迭代方式修改生成子[...]