Semi–preemptive routing on a line

The problem of routing a robot (or vehicle) between n stations in the plane in order to transport objects is well studied, even if the stations are specially arranged, e.g. on a linear track or circle. The robot may use either all or none of the stations for reloading. We will generalize these concepts of preemptiveness/non–preemptiveness and emancipate the robot by letting it choose k ⩽ n reload–stations.