# Two algorithms for the package-exchange robot-routing problem

## TL;DR

This paper introduces two algorithms, RIP and bubbletree, for the package-exchange robot-routing problem, providing theoretical bounds on their makespan performance and demonstrating their efficiency in different scenarios.

## Contribution

The paper presents two novel algorithms for PERR with proven performance bounds, advancing the theoretical understanding of robot-routing efficiency.

## Key findings

- RIP guarantees a makespan of O(SIC + k^2).
- Bubbletree achieves a makespan of O(n) and can be optimized to O((k+l)log k).
- Both algorithms improve theoretical bounds for PERR.

## Abstract

We present and analyze two new algorithms for the package-exchange robot-routing problem (PERR): restriction to inidividual paths (RIP) and bubbletree. RIP provably produces a makespan that is $O(\text{SIC}+k^2)$, where SIC is the sum of the lengths of the individual paths and $k$ is the number of robots. Bubbletree produces a makespan that is $O(n)$, where $n$ is the number of nodes. With optimizations bubbletree can also achieve a makespan of $O((k+l)\text{log}k)$, where $l$ is the longest path from start to goal in the bubbletree subgraph.