Approximation algorithms for two clustered arc routing problems

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.