Semi-Präemptives Transportieren

Abstract

We consider the problem of routing a robot (or vehicle) between n stations in order to transport m objects from their source node to their destination node such that the total travel distance is minimized. This problem is known as Pickup and Delivery problem and well studied in literature, 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. These cases are said to be preemptive resp. non--preemptive. We will generalize these concepts by restricting the nodes that can be used for reloading. According to the previous versions we will call this variation semi--preemptive. We will distinguish between an exogenous and an endogenous case. Problems of the first type will allow the robot to use only a subset of stations that can be used as reload nodes. The latter case consists of a number k and it will be part of the problem to decide at which stations the robot reloads, but no more than k times. We will show that both the exogenous and the endogenous problem are efficiently solvable by dynamic programming on paths and circles. Both variants are NP --complete on trees. For the exogenous case we will give an approximation algorithm with ratio 4/3 . The endogenous case can be solved with ratio (4/3+c) , for arbitrary c>0 . In conclusion we use a known result in order to approximately solve the exogenous resp. endogenous routing problem on graphs weighted by a metric. We will study a graph theoretical problem which arises from the mentioned transportation problem. Given a directed graph G=(V,E^rdotcup E^b) with the arc set partitioned into red and blue subsets and a cost function on the arc set, find a minimal cost arborescence spanning G and using at most d blue arcs. Within this work a fully polynomial time approximation scheme (FPTAS) will be presented for this problem. The FPTAS will be used to approximately solve the endogenous transportation problem on a tree