An integer programming formulation using convex polygons for the convex partition problem

TL;DR

This paper introduces a new integer programming approach for the convex partition problem, improving solution efficiency and scalability for larger point sets by leveraging geometric properties.

Contribution

The paper presents a novel IP formulation based on face representation, significantly enhancing previous models and enabling optimal solutions for larger datasets.

Findings

Easily solves 100-point datasets to optimality

Provides tight lower bounds for datasets up to 300 points

Outperforms existing formulations in efficiency and scalability

Abstract

A convex partition of a point set P in the plane is a planar partition of the convex hull of P with empty convex polygons or internal faces whose extreme points belong to P. In a convex partition, the union of the internal faces give the convex hull of P and the interiors of the polygons are pairwise disjoint. Moreover, no polygon is allowed to contain a point of P in its interior. The problem is to find a convex partition based on the minimum number of internal faces. The problem has been shown to be NP-Hard and was recently used in the CG:SHOP Challenge 2020. We propose a new integer linear programming (IP) formulation that considerably improves over the existing one. It relies on the representation of faces as opposed to segments and points. A number of geometric properties are used to strengthen it. Data sets of 100 points are easily solved to optimality and the lower bounds provided by the model can be computed up to 300 points.