SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 953-969, January 2021.
Given an $n$-vertex $m$-edge graph $G$ with nonnegative edge-weights, a shortest cycle is one minimizing the sum of the weights on its edges. The girth of $G$ is the weight of a shortest cycle. We obtain several new approximation algorithms for computing the girth of weighted graphs: For any graph $G$ with polynomially bounded integer weights, we present a deterministic algorithm that computes, in $\tilde{\cal O}(n^{5/3}+m)$-time, a cycle of weight at most twice the girth of $G$. This matches the approximation factor of the best known subquadratic-time approximation algorithm for the girth of unweighted graphs. Then, we turn our algorithm into a deterministic $(2+\varepsilon)$-approximation for graphs with arbitrary nonnegative edge-weights, at the price of a slightly worse running time in $\tilde{\cal O}(n^{5/3}\text{polylog}{(1/\varepsilon)}+m)$. For that, we introduce a generic method in order to obtain a polynomial-factor approximation of the girth in subquadratic time, that may be of independent interest. Finally, if we assume that the adjacency lists are sorted then we can get rid off the dependency in the number $m$ of edges. Namely, we can transform our algorithms into an $\tilde{\cal O}(n^{5/3})$-time randomized 4-approximation for graphs with nonnegative edge-weights. This can be derandomized, thereby leading to an $\tilde{\cal O}(n^{5/3})$-time deterministic 4-approximation for graphs with polynomially bounded integer weights, and an $\tilde{\cal O}(n^{5/3}\text{polylog}{(1/\varepsilon)})$-time deterministic $(4+\varepsilon)$-approximation for graphs with nonnegative edge-weights. To the best of our knowledge, these are the first known subquadratic-time approximation algorithms for computing the girth of weighted graphs.

中文翻译：

---

用于计算加权图上最短周期的更快逼近算法

SIAM 离散数学杂志，第 35 卷，第 2 期，第 953-969 页，2021 年 1 月。
给定具有非负边权重的 $n$-顶点 $m$-边图 $G$，最短循环是最小化其边上的权重总和的循环。$G$ 的周长是最短周期的重量。我们获得了几种用于计算加权图周长的新近似算法：对于具有多项式有界整数权重的任何图 $G$，我们提出了一种确定性算法，该算法计算 $\tilde{\cal O}(n^{5/3 }+m)$-time，重量最多为$G$周长的两倍。这与用于未加权图的周长的最著名的次二次时间近似算法的近似因子相匹配。然后，我们将我们的算法转化为具有任意非负边权重的图的确定性 $(2+\varepsilon)$-近似，代价是 $\tilde{\cal O}(n^{5/3}\text{polylog}{(1/\varepsilon)}+m)$ 中的运行时间稍差。为此，我们引入了一种通用方法，以获得亚二次时间中周长的多项式因子近似值，这可能是独立的兴趣。最后，如果我们假设邻接表是排序的，那么我们可以摆脱对 $m$ 边数的依赖。也就是说，对于具有非负边权重的图，我们可以将我们的算法转换为 $\tilde{\cal O}(n^{5/3})$-time 随机 4-approximation。这可以被去随机化，从而导致具有多项式有界整数权重的图的 $\tilde{\cal O}(n^{5/3})$-time 确定性 4-approximation，和 $\tilde{\cal O}(n^{5/3}\text{polylog}{(1/\varepsilon)})$-时间确定性 $(4+\varepsilon)$-近似于非负图边权重。据我们所知，这些是第一个已知的用于计算加权图周长的次二次时间近似算法。