A Closer Cut: Computing Near-Optimal Lawn Mowing Tours

TL;DR

This paper introduces a practical approach for computing near-optimal tours for the Lawn Mowing Problem, a complex geometric optimization task, achieving high-quality solutions efficiently for large instances.

Contribution

It provides the first practical primal-dual algorithm with convergence guarantees for the Lawn Mowing Problem, enabling near-optimal solutions for large instances.

Findings

Achieved provably optimal and near-optimal solutions for instances with up to 2000 vertices.

Developed a primal-dual method based on TSPN for practical computation.

Theoretical insights show optimal solutions are polygonal paths with bounded vertices.

Abstract

For a given polygonal region $P$, the Lawn Mowing Problem (LMP) asks for a shortest tour $T$ that gets within Euclidean distance 1 of every point in $P$; this is equivalent to computing a shortest tour for a unit-disk cutter $C$ that covers all of $P$. As a geometric optimization problem of natural practical and theoretical importance, the LMP generalizes and combines several notoriously difficult problems, including minimum covering by disks, the Traveling Salesman Problem with neighborhoods (TSPN), and the Art Gallery Problem (AGP). In this paper, we conduct the first study of the Lawn Mowing Problem with a focus on practical computation of near-optimal solutions. We provide new theoretical insights: Optimal solutions are polygonal paths with a bounded number of vertices, allowing a restriction to straight-line solutions; on the other hand, there can be relatively simple instances for which optimal solutions require a large class of irrational coordinates. On the practical side, we present a primal-dual approach with provable convergence properties based on solving a special case of the TSPN restricted to witness sets. In each iteration, this establishes both a valid solution and a valid lower bound, and thereby a bound on the remaining optimality gap. As we demonstrate in an extensive computational study, this allows us to achieve provably optimal and near-optimal solutions for a large spectrum of benchmark instances with up to 2000 vertices.