AI 中文总结
提出加权图最短环（周长）的近似算法，实现与无权图相同的逼近比与运行时间权衡，并证明无权图周长逼近的细粒度下界。
AI 中文摘要
我们研究给定图中最短环长度（即图的周长）的逼近问题。Kadria 等人 [SODA'22] 和 Roditty 与 Trabelsi [arXiv'25] 针对无权图的最新逼近算法实现了如下权衡：对于每个整数 $k\geq 2$，存在一个 $\tilde{O}(n^{1+2/k})$ 时间的算法，在 $n$ 节点无权图中实现周长的 $(2k/3)$-逼近。本文的第一个结果是在具有非负实数边权的 $m$ 边、$n$ 节点图中实现相同的权衡：一个 $2k/3$-逼近算法，运行时间为 $\tilde{O}(m+n^{1+2/k})$。在加权图中，对 $m$ 的依赖是不可避免的。我们的结果改进了 Kadria 等人 [SODA'23] 和 Ducoffe [ICALP'19 和 SIDMA'21] 的工作，他们仅能对某些 $k$ 值实现这种权衡。我们还证明了无权图中周长逼近及相关问题的新细粒度下界。
英文摘要
We study the problem of approximating the length of a shortest cycle in a given graph, known as the girth of the graph. The state-of-the-art approximation algorithms for unweighted graphs by Kadria et al. [SODA'22] and Roditty and Trabelsi [arXiv'25] achieve the following trade-off: for every integer $k\geq 2$, there is an $\tilde{O}(n^{1+2/k})$ time algorithm that achieves a $(2k/3)$-approximation for the girth in unweighted $n$-node graphs. The first result of this paper is to achieve the same trade-off for $m$-edge, $n$-node graphs with non-negative real edge weights: a $2k/3$-approximation algorithm running in $\tilde{O}(m+n^{1+2/k})$ time. The dependence on $m$ is unavoidable in weighted graphs. Our result improves on the work of Kadria et al.~[SODA'23] and Ducoffe [ICALP'19 and SIDMA'21], who were only able to achieve such a trade-off for some values of $k$. We also prove new fine-grained lower bounds for girth approximation and related problems in unweighted graphs.
