We present several results in the CONGEST model on round complexity for Replacement Paths (RPaths), Minimum Weight Cycle (MWC), and All Nodes Shortest Cycles (ANSC). We study these fundamental problems in both directed and undirected graphs, both weighted and unweighted. Many of our results are optimal to within a polylog factor: For an $n$-node graph $G$ we establish near linear lower and upper bounds for computing RPaths if $G$ is directed and weighted, and for computing MWC and ANSC if $G$ is weighted, directed or undirected; near $\sqrt{n}$ lower and upper bounds for undirected weighted RPaths; and $\Theta(D)$ bound for undirected unweighted RPaths. We also present lower and upper bounds for approximation versions of these problems, notably a $(2-(1/g))$-approximation algorithm for undirected unweighted MWC that runs in $\tilde{O}(\sqrt{n}+D)$ rounds, improving on the previous best bound of $\tilde{O}(\sqrt{ng}+D)$ rounds, where $g$ is the MWC length. We present a $(1+\epsilon)$-approximation algorithm for directed weighted RPaths, which beats the linear lower bound for exact RPaths.

翻译：我们在CONGEST模型中针对替换路径（RPaths）、最小权重环（MWC）和所有节点最短环（ANSC）问题的轮复杂度方面给出若干结果。我们在有向图和无向图、加权图和非加权图中研究了这些基本问题。许多结果在多项式对数因子内达到最优：对于包含$n$个节点的图$G$，若$G$为有向加权图，我们给出计算RPaths的近线性下界和上界；若$G$为加权图（有向或无向），则给出计算MWC和ANSC的近线性下界和上界；为无向加权RPaths给出近$\sqrt{n}$的下界和上界；为无向非加权RPaths给出$\Theta(D)$的界。我们还针对这些问题的近似版本给出下界和上界，特别提出一种针对无向非加权MWC的$(2-(1/g))$近似算法，该算法运行于$\tilde{O}(\sqrt{n}+D)$轮，改进了此前$\tilde{O}(\sqrt{ng}+D)$轮的最佳结果（其中$g$为MWC长度）。针对有向加权RPaths，我们提出一种$(1+\epsilon)$近似算法，该算法突破了精确RPaths的线性下界。