Optimal area polygonisation problems: Mixed integer linear programming models

Abstract

This paper describes approaches to finding polygons with maximum and minimum area through a given set of points on the Euclidean plane. Both problems are NP-hard and of interest in Computational Geometry. They have been extensively studied, although only the recent literature presents mathematical formulations to consistently find optimal solutions to instances from a benchmark collection with up to 25 points. We present and compare eight new Mixed Integer Linear Programming models for the two problems. Four represent a polygon as a solution to the Asymmetric Travelling Salesman Problem, while the other four use the Symmetric Travelling Salesman Problem representation. Six of the models select triangles to compute the area of a polygon, while the other two models select subpolygons. A computational analysis on the benchmark collection shows that one of our new formulations can consistently solve most of the benchmark instances with up to 50 points. The success of our proposal is a linear system to select a triangulation of the point set, a partition of the triangulation to cover the internal and external area of a polygon, and a representation of the polygon as the solution of an Asymmetric Travelling Salesman Problem.