On the time complexity of conflict-free vehicle routing

In this paper, we study the following problem: given n vehicles and origin-destination pairs in the plane, what is the minimum time needed to transfer each vehicle from its origin to its destination, avoiding conflicts with other vehicles? The environment is free of obstacles, and a conflict occurs when the distance between any two vehicles is smaller than a velocity-dependent safety distance. We derive lower and upper bounds on the time needed to complete the transfer, in the case in which the origin and destination points can be chosen arbitrarily, proving that the transfer takes θ(√nL~) time to complete, where L~ is the average distance between origins and destinations. We also analyze the case in which origin and destination points are generated randomly according to a uniform distribution, and present an algorithm providing a constructive upper bound on the time needed for complete the transfer, proving that in the random case the transfer requires O(√(n logn)) time.