Vehicle scheduling on a tree with release and handling times

Abstract

Let T=(V, E,v ₀) be a rooted tree, where V is a set of n vertices, E is a set of edges and v ₀ ∃ V is the root. The travel times d(v _i,v_j) and d(v _j,v_i) are associated with each edge (v _i,v_j) ∃ E, and a job, which is also denoted as v _i, is located at each vertex v _i. Each job v _i has release time r(v _j) and handling time h(v _i). The TREE-VSP (Vehicle Scheduling Problem on a Tree) asks to find a routing schedule of the vehicle such that it starts from root v ₀, visits all jobs v _i ∃ V for processing, and returns to v ₀. The processing of a job v _i cannot be started before its release time t=r(v _i) (hence the vehicle may have to wait if it arrives at v _i too early) and takes h(v _i) time units once its processing has been started (no interruption is allowed). The objective is to find a schedule that minimizes the completion time (i.e., the time to return to v ₀ after processing all jobs). We first prove that TREE-VSP is NP-hard. Then we show that TREE-VSP with depth-first routing constraint can be exactly solved in θ(n log n) time. Finally we show that, if we regard this exact algorithm as an approximate algorithm for TREE-VSP without such routing constraint, its worst-case ratio is at most two, and that this bound is tight.