Approximation algorithms for general cluster routing problem

Graph routing problems have been investigated extensively in operations research, computer science and engineering due to their ubiquity and vast applications. In this paper, we study constant approximation algorithms for some variations of the general cluster routing problem. In this problem, we are given an edge-weighted complete undirected graph $G=(V,E,c),$ whose vertex set is partitioned into clusters $C_1,\dots ,C_k\mathrm{.}$ We are also given a subset $V^{\mathrm{\prime }}$ of $V$ and a subset $E^{\mathrm{\prime }}$ of $E\mathrm{.}$ The weight function $c$ satisfies the triangle inequality. The goal is to find a minimum cost walk $T$ that visits each vertex in $V^{\mathrm{\prime }}$ only once, traverses every edge in $E^{\mathrm{\prime }}$ at least once and for every $i\in [k]$ all vertices of $C_i$ are traversed consecutively.