We investigate the \emph{minimum weight cycle (MWC)} problem in the $\mathsf{CONGEST}$ model of distributed computing. For undirected weighted graphs, we design a randomized algorithm that achieves a $(k+1)$-approximation, for any \emph{real} number $k \ge 1$. The round complexity of algorithm is \[ \tilde{O}\!\Big( n^{\frac{k+1}{2k+1}} + n^{\frac{1}{k}} + D\, n^{\frac{1}{2(2k+1)}} + D^{\frac{2}{5}} n^{\frac{2}{5}+\frac{1}{2(2k+1)}} \Big). \] where $n$ denotes the number of nodes and $D$ is the unweighted diameter of the graph. This result yields a smooth trade-off between approximation ratio and round complexity. In particular, when $k \geq 2$ and $D = \tilde{O}(n^{1/4})$, the bound simplifies to \[ \tilde{O}\!\left( n^{\frac{k+1}{2k+1}} \right) \] On the lower bound side, assuming the Erdős girth conjecture, we prove that for every \emph{integer} $k \ge 1$, any randomized $(k+1-ε)$-approximation algorithm for MWC requires \[ \tildeΩ\!\left( n^{\frac{k+1}{2k+1}} \right) \] rounds. This lower bound holds for both directed unweighted and undirected weighted graphs, and applies even to graphs with small diameter $D = Θ(\log n)$. Taken together, our upper and lower bounds \emph{match up to polylogarithmic factors} for graphs of sufficiently small diameter $D = \tilde{O}(n^{1/4})$ (when $k \geq 2$), yielding a nearly tight bound on the distributed complexity of the problem. Our results improve upon the previous state of the art: Manoharan and Ramachandran (PODC~2024) demonstrated a $(2+ε)$-approximation algorithm for undirected weighted graphs with round complexity $\tilde{O}(n^{2/3}+D)$, and proved that for any arbitrarily large number $α$, any $α$-approximation algorithm for directed unweighted or undirected weighted graphs requires $Ω(\sqrt{n}/\log n)$ rounds.

翻译：我们研究分布式计算$\mathsf{CONGEST}$模型中的\textbf{最小权重环（MWC）}问题。对于无向加权图，我们设计了一种随机化算法，针对任意\textit{实数}$k \ge 1$可实现$(k+1)$-近似。该算法的轮复杂度为\[ \tilde{O}\!\Big( n^{\frac{k+1}{2k+1}} + n^{\frac{1}{k}} + D\, n^{\frac{1}{2(2k+1)}} + D^{\frac{2}{5}} n^{\frac{2}{5}+\frac{1}{2(2k+1)}} \Big), \]其中$n$表示节点数，$D$为图的未加权直径。该结果在近似比与轮复杂度之间建立了平滑权衡。特别地，当$k \geq 2$且$D = \tilde{O}(n^{1/4})$时，复杂度简化为\[ \tilde{O}\!\left( n^{\frac{k+1}{2k+1}} \right). \]在下界方面，假设Erdős周长猜想成立，我们证明：对于每个\textit{整数}$k \ge 1$，任意随机$(k+1-ε)$-近似MWC算法至少需要\[ \tildeΩ\!\left( n^{\frac{k+1}{2k+1}} \right) \]轮。该下界同时适用于有向无权图和无向加权图，且对直径$D = Θ(\log n)$的小直径图仍然成立。综合来看，对于足够小直径$D = \tilde{O}(n^{1/4})$的图（当$k \geq 2$时），我们的上下界\textit{在多项式对数因子范围内匹配}，给出了该问题几乎紧的分布式复杂界限。我们的结果改进了先前最优成果：Manoharan和Ramachandran（PODC 2024）针对无向加权图给出了轮复杂度为$\tilde{O}(n^{2/3}+D)$的$(2+ε)$-近似算法，并证明对于任意充分大的数$α$，针对有向无权图或无向加权图的任意$α$-近似算法至少需要$Ω(\sqrt{n}/\log n)$轮。