An Efficient Algorithm for the Fast Delivery Problem Iago A. Carvalho 1[0000−0001−9404−1329] , Thomas Erlebach 2[0000−0002−4470−5868] , and Kleitos Papadopoulos 2[0000−0002−7086−0335] 1 Department of Computer Science, Universidade Federal de Minas Gerais, Brazil iagoac@dcc.ufmg.br, https://iagoac.github.io 2 Department of Informatics, University of Leicester, England, UK te17@leicester.ac.uk, kleitospa@gmail.com Abstract. We study a problem where k autonomous mobile agents are initially located on distinct nodes of a weighted graph (with n nodes and m edges). Each autonomous mobile agent has a predefined velocity and is only allowed to move along the edges of the graph. We are interested in delivering a package, initially positioned in a source node s, to a des- tination node y. The delivery is achieved by the collective effort of the autonomous mobile agents, which can carry and exchange the package among them. The objective is to compute a delivery schedule that min- imizes the delivery time of the package. In this paper, we propose an O(kn log(kn)+ km) time algorithm for this problem. This improves the previous state-of-the-art O(k 2 m + kn 2 + APSP) time algorithm for this problem, where APSP stands for the running-time of an algorithm for the All-Pairs Shortest Paths problem. Keywords: Mobile agents · Dijkstra’s algorithm · Polynomial-time al- gorithm · Time-dependent shortest paths 1 Introduction Enterprises, such as DHL, UPS, Swiss Post, and Amazon, are now delivering goods and packages to their clients using autonomous drones [1,17]. Those drones depart from a base (which can be static, such as a warehouse [15], or mobile, such as a truck or a van [16]) and deliver the package into their clients’ houses or in the street. However, packages are not delivered to a client that is too far from the drone’s base due to the energy limitations of such autonomous aerial vehicles. In the literature, we find some proposals for delivering packages over a longer distance. One of them, proposed by Hong, Kuby, and Murray [15], is to install recharging bases in several spots, which allows a drone to stop, recharge, and continue its path. However, this strategy may result in a delayed delivery, because drones may stop several times to recharge during a single delivery. A manner to overcome this limitation is to use a swarm of drones. The idea of this technique is to position drones in recharging bases all over the delivery