Euler is standing in line dial-a-ride problems with precedence-constraints Original Research Article

In this paper we study algorithms for “Dial-a-Ride” transportation problems. In the basic version of the problem we are given transportation jobs between the vertices of a graph and the goal is to find a shortest transportation that serves all the jobs. This problem is known to be NP-hard even on trees. We consider the extension when precedence relations between the jobs with the same source are given. Our results include a polynomial time algorithm on paths and approximation algorithms for general graphs and trees with performances of 9/4 and 5/3, respectively.