The Multiple Traveling Salesman Problem on Spiders.

Given (i) a set of N + 1 vertices, that corresponds to N clients and 1 depot, (ii) the travel time between each pair of vertices and (iii) a number m of salespersons, the multiple traveling salesman problem consists in finding m tours such that, starting from the depot, all clients are visited in such a way that some objective function is minimized. The objective function we consider in this paper is the makespan. More precisely, the goal is to find m tours (one for each salesperson) that minimize the time elapsed from the beginning of the operation until the last salesman comes back to the depot. We take into account handling times, i.e., the time spent visiting each client, which we assume to be the same for all of them. We address the problem in the particular case where the depot-clients network is a spider, with the depot located at its center. We show that this case is NP-hard even for 2 salespersons. We also show structural properties of the optimal solutions. These properties allow us to devise a PTAS for minimizing the makespan. More precisely, a (1+ epsilon)-approximation algorithm with running time N-O(m/epsilon).