# Clustering in Discrete Path Planning for Approximating Minimum Length Paths

## TL;DR

This paper introduces Gamma-Clustering, a method for discrete robot path planning on graphs that reduces computational complexity while maintaining near-optimal solutions, demonstrated through traveling salesman problem instances.

## Contribution

The paper presents a novel clustering approach for graph-based path planning that balances solution quality and search space reduction, with an efficient algorithm for optimal clustering.

## Key findings

- Significant reduction in computation time on TSP instances
- Solutions remain close to optimal despite clustering
- Optimal Gamma-Clusters are unique and efficiently found

## Abstract

In this paper we consider discrete robot path planning problems on metric graphs. We propose a clustering method, Gamma-Clustering for the planning graph that significantly reduces the number of feasible solutions, yet retains a solution within a constant factor of the optimal. By increasing the input parameter Gamma, the constant factor can be decreased, but with less reduction in the search space. We provide a simple polynomial- time algorithm for finding optimal Gamma-Clusters, and show that for a given Gamma, this optimal is unique. We demonstrate the effectiveness of the clustering method on traveling salesman instances, showing that for many instances we obtain significant reductions in computation time with little to no reduction in solution quality.