# An Improved Upper Bound for the Ring Loading Problem

## TL;DR

This paper improves the theoretical upper bound on the maximum load in the ring loading problem, narrowing the gap between known bounds and advancing understanding of optimal routing solutions.

## Contribution

It establishes a new upper bound of L ≤ L* + 1.3D for the ring loading problem, refining previous bounds and contributing to the theoretical analysis of the problem.

## Key findings

- New upper bound L ≤ L* + 1.3D established
- Improves upon previous bounds by Schrijver et al. and Skutella
- Provides initial bounds for small instances

## Abstract

The Ring Loading Problem emerged in the 1990s to model an important special case of telecommunication networks (SONET rings) which gained attention from practitioners and theorists alike. Given an undirected cycle on $n$ nodes together with non-negative demands between any pair of nodes, the Ring Loading Problem asks for an unsplittable routing of the demands such that the maximum cumulated demand on any edge is minimized. Let $L$ be the value of such a solution. In the relaxed version of the problem, each demand can be split into two parts where the first part is routed clockwise while the second part is routed counter-clockwise. Denote with $L^*$ the maximum load of a minimum split routing solution. In a landmark paper, Schrijver, Seymour and Winkler [SSW98] showed that $L \leq L^* + 1.5D$, where $D$ is the maximum demand value. They also found (implicitly) an instance of the Ring Loading Problem with $L = L^* + 1.01D$. Recently, Skutella [Sku16] improved these bounds by showing that $L \leq L^* + \frac{19}{14}D$, and there exists an instance with $L = L^* + 1.1D$. We contribute to this line of research by showing that $L \leq L^* + 1.3D$. We also take a first step towards lower and upper bounds for small instances.