Vehicle scheduling on a tree with release and handling times
Let T=(V, E,v0) be a rooted tree, where V is a set of n vertices, E is a set of edges and v0 ∃ V is the root. The travel times d(vi,vj) and d(vj,vi) are associated with each edge (vi,vj) ∃ E, and a job, which is also denoted as vi, is located at each vertex vi. Each job vi has release time r(vj) and handling time h(vi). The TREE-VSP (Vehicle Scheduling Problem on a Tree) asks to find a routing schedule of the vehicle such that it starts from root v0, visits all jobs vi ∃ V for processing, and returns to v0. The processing of a job vi cannot be started before its release time t=r(vi) (hence the vehicle may have to wait if it arrives at vi too early) and takes h(vi) 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 v0 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.
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