On Euclidean vehicle routing with allocation

Abstract: The (Euclidean) Vehicle Routing Allocation Problem (VRAP) is a generalization of Euclidean TSP. We do not require that all points lie on the salesman tour. However, points that do not lie on the tour are allocated, i.e., they are directly connected to the nearest tour point, paying a higher (per-unit) cost. More formally, the input is a set of n points and functions and . We wish to compute a subset and a salesman tour π through T such that the total length of the tour plus the total allocation cost is minimum. The allocation cost for a single point is , where is the nearest point on the tour. We give a PTAS with complexity for this problem. Moreover, we propose an -time PTAS for the Steiner variant of this problem. This dramatically improves a recent result of Armon et al. [A. Armon, A. Avidor, O. Schwartz, Cooperative TSP, in: Proceedings of the 14th Annual European Symposium on Algorithms, 2006, pp. 40–51]. [Copyright &y& Elsevier]