Vehicle scheduling on a tree with release and handling times

Abstract

In this paper, we consider a single-vehicle scheduling problem on a tree-shaped road net-work. Let T =(V,E) be a tree, where V is a set of n vertices and E is a set of edges. A task is located at each vertex v, which is also denoted as v. Each task v has release time r(v) and handling time h(v). The travel times c(u, v) and c(v, u) are associated with each edge (u, v) of E. The vehicle starts from an initial vertex v_0 of V, visits all tasks v in V for their processing, and returns to v_0 . The objective is to find a routing schedule of the vehicle that minimizes the completion time, denoted as C (i.e., the time to return to v_0 after processing all tasks). We call this problem TREE-VSP(C). We first prove that TREE-VSP(C) is NP-hard. However, we then show that TREE-VSP(C) with a depth-first routing constraint can be exactly solved in O(n log n) time. Moreover, we show that, if this exact algorithm is used as an approximate algorithm for the original TREE-VSP(C), its worst-case performance ratio is at most two.