Solving the single-vehicle scheduling problems for all home locations under depth-first routing on a tree

We consider a single-vehicle scheduling problem on a tree, where each verte x has a job with a release time and a processing time and each edge has a t ravel time. There is a single vehicle which starts from a start vertex s an d reaches a goal vertex g after finishing all jobs. In particular, s is cal led a home location if s = g. The objective of the problem is to find a dep th-first routing on T so as to minimize the completion time. In this paper, we first show that the minimum completion times of the problem for all hom e locations s ' is an element of V can be simultaneously computed in O(n) t ime, once the problem with a specified home location s ' is an element of V has been solved, where n is the number of vertices. We also show that give n a specified start vertex s, the minimum completion times for all goal ver tices g can be computed in O(n) time.