Given a strongly connected mixed graph \(G=(V,E,A)\), where V represents the vertex set, E is the undirected edge set, and A is the directed arc set, \(R \subseteq E\) is a subset of required edges and is divided into p clusters \(R_1,R_2,\dots ,R_p\), and A is a set of required arcs and is partitioned into q clusters \(A_1,A_2,\ldots ,A_q\). Each edge in E and each arc in A are associated with a nonnegative weight and the weight function satisfies the triangle inequality. In this paper we consider two clustered arc routing problems. The first is the Clustered Rural Postman Problem, in which A is empty and the objective is to find a minimum-weight closed walk such that all the edges in R are serviced and the edges in \(R_i\) (\(1\le i \le p\)) are serviced consecutively. The other is the Clustered Stacker Crane Problem, in which R is empty and the goal is to find a minimum-weight closed walk that traverses all the arcs in A and services the arcs in \(A_j\) (\(1\le j \le q\)) consecutively. For both problems, we propose constant-factor approximation algorithms with ratios 13/6 and 19/6, respectively.

给定一个强连通混合图\(G=(V,E,A)\)，其中V表示顶点集，E是无向边集，A是有向弧集，\(R\subseteq E\)是所需边的子集，分为 p 个簇 \(R_1,R_2,\dots ,R_p\)，A 是所需弧的集合，分为 q 个簇 \(A_1,A_2,\ldots ,A_q\ ）。 E中的每条边和A中的每条弧都与一个非负权重相关联，并且权重函数满足三角不等式。在本文中，我们考虑两个集群弧路由问题。第一个是集群农村邮递员问题，其中 A 为空，目标是找到一条最小权重封闭游走，使得 R 中的所有边都得到服务，且 \(R_i\) (\(1\le i \le p\)) 连续提供服务。另一个是集群堆垛机问题，其中 R 为空，目标是找到一条遍历 A 中所有弧并服务于 \(A_j\) (\(1\le j \le q\)) 连续。对于这两个问题，我们分别提出了比率为 13/6 和 19/6 的常数因子近似算法。