# An Efficient Algorithm for the Fast Delivery Problem

## TL;DR

This paper introduces a new efficient algorithm for the fast delivery problem involving autonomous agents on a graph, significantly reducing the computation time compared to previous methods.

## Contribution

The paper presents an O(kn log n + km) time algorithm for package delivery with autonomous agents, improving upon the previous state-of-the-art algorithms.

## Key findings

- The new algorithm reduces the delivery scheduling computation time.
- It outperforms previous algorithms in terms of efficiency.
- The approach is applicable to large-scale graph-based delivery problems.

## 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 destination 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 minimizes the delivery time of the package. In this paper, we propose an O(kn log n + km) time algorithm for this problem. This improves the previous state-of-the-art O(k^2 m + k n^2 + APSP) time algorithm for this problem, where APSP stands for the running-time of an algorithm for the All-Pairs Shortest Paths problem.