ISAAC 1993: Algorithms and Computation pp 486-495

# Vehicle scheduling on a tree with release and handling times

• Yoshiyuki Karuno
• Hiroshi Nagamochi
• Toshihide Ibaraki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 762)

## Abstract

Let T=(V, E,v0) be a rooted tree, where V is a set of n vertices, E is a set of edges and v0V 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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© Springer-Verlag Berlin Heidelberg 1993

## Authors and Affiliations

• Yoshiyuki Karuno
• 1
• Hiroshi Nagamochi
• 2
• Toshihide Ibaraki
• 2
1. 1.Department of Mechanical and System EngineeringKyoto Institute of TechnologyJapan
2. 2.Department of Applied Mathematics and PhysicsKyoto UniversityJapan