Collaborative Delivery on a Fixed Path with Homogeneous Energy-Constrained Agents

Abstract

We consider the problem of collectively delivering a package from a specified source to a designated target location in a graph, using multiple mobile agents. Each agent starts from some vertex of the graph; it can move along the edges of the graph and can can pick up the package from a vertex and drop it in another vertex during the course of its movement. However, each agent has limited energy budget allowing it to traverse a path of bounded length B; thus, multiple agents need to collaborate to move the package to its destination. Given the positions of the agents in the graph and their energy budgets, the problem of finding a feasible movement schedule is called the Collaborative Delivery problem and has been studied before.One of the open questions from previous results is what happens when the delivery must follow a fixed path given in advance. Although this special constraint reduces the search space for feasible solutions, we show that the problem of finding a feasible schedule remains NP hard (as the original problem). We consider the optimization version of the problem that asks for the optimal energy budget B per agent which allows for a feasible delivery schedule, given the initial positions of the agents. We show the existence of better approximations for the fixed-path version of the problem (at least for the restricted case of a single pickup per agent), compared to the known results for the general version of the problem.We provide polynomial time approximation algorithms for both directed and $ This work was partially supported by the ANR project ANCOR (anr-14-CE36-0002-01).