Department of Applied Mathematics & Physics, Kyoto Univiversity
Technical Report #97008 (October, 1997)
Single-Vehicle Scheduling Problem on a Tree with Depth-First Routing
by Hiroshi Nagamochi, Koji Mochizuki and Toshihide Ibaraki
We consider a single-vehicle scheduling problem on a tree.
Let $T=(V,E)$ be a tree. The travel times $d(u,v)$ and $d(v,u)$
are associated with each edge
$\{u,v\}\in E$, and a job, which is also denoted as $v$, is located at each
vertex $v\in V$.
Each job $v$ has release time $r(v)$ and handling time $h(v)$.
There is a single vehicle, which is initially situated
at the start vertex $s\in V$, and visits all the jobs in $T$ for processing,
before reaching the goal vertex $g\in V$.
The start vertex $s$ is called the home location
if it is also the goal vertex $g$.
The processing of a job $v$ cannot be started before its release time
$t=r(v)$ and takes $h(v)$ time units once its processing has started.
We also add the depth-first routing constraints;
i.e., the vehicle has to complete all jobs in the subtree rooted at
every vertex $v$ before returning to its parent vertex.
The resulting problem of minimizing the completion time
under the depth-first routing constraint is denoted by TREE-VSP/DF.
In this paper, we first show that the TREE-VSP/DF with a home location $s$
has an optimal schedule that is
independent of its start time, and that such schedule
can be found in $O(n\log \Delta)$ time,
where $\Delta$ is the maximum degree of tree $T$
and $n$ is the number of vertices.
We then show that the minimum completion times $C(s')$
for all home locations $s'\in V$ can be simultaneously
computed in $O(n)$ time, once the problem
with a specified home location has been solved.
We also consider the TREE-VSP/DF with a start vertex $s$
and a goal vertex $g$, and show that, once the TREE-VSP/DF with a specified
home location $s$ has been solved, the minimum completion times $C(s,g)$
for all goal vertices $g\in V$ can be computed in $O(n)$ time.