Augmented Graphs of Convex Sets and the Traveling Salesman Problem

TL;DR

This paper introduces a novel trajectory optimization algorithm for the TSP in graphs of convex sets, transforming it into a shortest path problem and providing scalable heuristics with performance certification.

Contribution

It develops an augmented graph framework for exact TSP solutions in GCS and proposes a branch and bound heuristic for larger instances.

Findings

Exact solution framework for TSP in GCS using shortest path formulation

A branch and bound heuristic with minimum 1-trees scales to larger problems

Exploration of lower bounds for performance certification

Abstract

We present a trajectory optimization algorithm for the traveling salesman problem (TSP) in graphs of convex sets (GCS). Our framework uses an augmented graph of convex sets to encode the TSP specification and solve it exactly as a shortest path problem in GCS. We establish a precise relationship between the landmark Bellman-Held-Karp algorithm and the augmented graph of convex sets with a TSP specification. Additionally, we present a branch and bound heuristic that uses minimum 1-trees to obtain certifiably optimal or near optimal solutions and scales to problems far larger than the exact framework can handle. To assess and certify performance, we explore several alternative lower bounds.