A Closer Cut: Computing Near-Optimal Lawn Mowing Tours

11/10/2022
by   Sandor P. Fekete, et al.
0

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.

READ FULL TEXT

page 4

page 5

page 12

page 16

page 17

research
07/03/2023

The Lawn Mowing Problem: From Algebra to Algorithms

For a given polygonal region P, the Lawn Mowing Problem (LMP) asks for a...
research
10/02/2020

Modeling all alternative solutions for highly renewable energy systems

As the world is transitioning towards highly renewable energy systems, a...
research
02/18/2018

Solving Large-Scale Minimum-Weight Triangulation Instances to Provable Optimality

We consider practical methods for the problem of finding a minimum-weigh...
research
04/02/2021

Branch-and-cut algorithms for the covering salesman problem

The Covering Salesman Problem (CSP) is a generalization of the Traveling...
research
08/23/2018

One (more) line on the most Ancient Algorithm in History

We give a new simple and short ("one-line") analysis for the runtime of ...
research
12/17/2018

Optimality Clue for Graph Coloring Problem

In this paper, we present a new approach which qualifies or not a soluti...
research
10/22/2021

A Simple Boosting Framework for Transshipment

Transshipment, also known under the names of earth mover's distance, unc...

Please sign up or login with your details

Forgot password? Click here to reset