An $(\alpha,\beta)$-spanner of a weighted graph $G=(V,E)$, is a subgraph $H$ such that for every $u,v\in V$, $d_G(u,v) \le d_H(u,v)\le\alpha\cdot d_G(u,v)+\beta$. The main parameters of interest for spanners are their size (number of edges) and their lightness (the ratio between the total weight of $H$ to the weight of a minimum spanning tree). In this paper we focus on near-additive spanners, where $\alpha=1+\varepsilon$ for arbitrarily small $\varepsilon>0$. We show the first construction of {\em light} spanners in this setting. Specifically, for any integer parameter $k\ge 1$, we obtain an $(1+\varepsilon,O(k/\varepsilon)^k\cdot W(\cdot,\cdot))$-spanner with lightness $\tilde{O}(n^{1/k})$ (where $W(\cdot,\cdot)$ indicates for every pair $u, v \in V$ the heaviest edge in some shortest path between $u,v$). In addition, we can also bound the number of edges in our spanner by $O(kn^{1+3/k})$.

翻译：加权图 $G=(V,E)$ 的 $(\alpha,\beta)$-生成器是其子图 $H$，使得对于任意 $u,v\in V$，满足 $d_G(u,v) \le d_H(u,v)\le\alpha\cdot d_G(u,v)+\beta$。生成器的主要关注参数是其规模（边数）和轻量性（$H$ 的总权重与最小生成树权重之比）。本文聚焦于近可加性生成器，其中 $\alpha=1+\varepsilon$ 且 $\varepsilon>0$ 可任意小。我们首次在此设定下构建了{\em 轻量}生成器。具体而言，对于任意整数参数 $k\ge 1$，我们获得了 $(1+\varepsilon,O(k/\varepsilon)^k\cdot W(\cdot,\cdot))$-生成器，其轻量性为 $\tilde{O}(n^{1/k})$（其中 $W(\cdot,\cdot)$ 表示对任意顶点对 $u, v \in V$，在 $u,v$ 间某条最短路径中最重的边）。此外，我们还能将生成器的边数限制为 $O(kn^{1+3/k})$。