The greedy algorithm adapted from Kruskal's algorithm is an efficient and folklore way to produce a $k$-spanner with girth at least $k+2$. The greedy algorithm has shown to be `existentially optimal', while it's not `universally optimal' for any constant $k$. Here, `universal optimality' means an algorithm can produce the smallest $k$-spanner $H$ given any $n$-vertex input graph $G$. However, how well the greedy algorithm works compared to `universal optimality' is still unclear for superconstant $k:=k(n)$. In this paper, we aim to give a new and fine-grained analysis of this problem in undirected unweighted graph setting. Specifically, we show some bounds on this problem including the following two (1) On the negative side, when $k<\frac{1}{3}n-O(1)$, the greedy algorithm is not `universally optimal'. (2) On the positive side, when $k>\frac{2}{3}n+O(1)$, the greedy algorithm is `universally optimal'. We also introduce an appropriate notion for `approximately universal optimality'. An algorithm is $(\alpha,\beta)$-universally optimal iff given any $n$-vertex input graph $G$, it can produce a $k$-spanner $H$ of $G$ with size $|H|\leq n+\alpha(|H^*|-n)+\beta$, where $H^*$ is the smallest $k$-spanner of $G$. We show the following positive bounds. (1) When $k>\frac{4}{7}n+O(1)$, the greedy algorithm is $(2,O(1))$-universally optimal. (2) When $k>\frac{12}{23}n+O(1)$, the greedy algorithm is $(18,O(1))$-universally optimal. (3) When $k>\frac{1}{2}n+O(1)$, the greedy algorithm is $(32,O(1))$-universally optimal. All our proofs are constructive building on new structural analysis on spanners. We give some ideas about how to break small cycles in a spanner to increase the girth. These ideas may help us to understand the relation between girth and spanners.

翻译：改编自Kruskal算法的贪婪算法是一种高效且广为人知的方法，用于生成周长至少为$k+2$的$k$-生成子图。该算法已被证明具有“存在最优性”，但对于任意常数$k$均不具备“通用最优性”。此处“通用最优性”指算法能在给定任意$n$顶点输入图$G$时生成最小的$k$-生成子图$H$。然而，对于超常数$k:=k(n)$，贪婪算法相较于“通用最优性”的表现仍不明确。本文旨在无向无权图设定下对该问题给出新的细粒度分析。具体而言，我们证明了该问题的若干界限，包括以下两点：(1) 负面结果：当$k<\frac{1}{3}n-O(1)$时，贪婪算法不具备“通用最优性”。(2) 正面结果：当$k>\frac{2}{3}n+O(1)$时，贪婪算法具有“通用最优性”。我们还引入了“近似通用最优性”的恰当定义：若算法在给定任意$n$顶点输入图$G$时，能生成满足$|H|\leq n+\alpha(|H^*|-n)+\beta$的$k$-生成子图$H$（其中$H^*$为$G$的最小$k$-生成子图），则称该算法具有$(\alpha,\beta)$-通用最优性。我们证明了以下正面界限：(1) 当$k>\frac{4}{7}n+O(1)$时，贪婪算法具有$(2,O(1))$-通用最优性。(2) 当$k>\frac{12}{23}n+O(1)$时，贪婪算法具有$(18,O(1))$-通用最优性。(3) 当$k>\frac{1}{2}n+O(1)$时，贪婪算法具有$(32,O(1))$-通用最优性。所有证明均基于对生成子图的新颖结构分析并具有构造性。我们提出了若干通过破坏生成子图中的小环来增加周长的思路，这些思路可能有助于理解周长与生成子图之间的关系。