Computing Area-Optimal Simple Polygonizations

TL;DR

This paper develops exact methods combining geometry and integer programming to find area-optimal simple polygonizations of point sets, solving instances of up to 25 points to proven optimality, highlighting the problem's computational difficulty.

Contribution

It introduces the first exact methods for Min-Area and Max-Area simple polygonizations, extending solvable instance sizes and demonstrating practical NP-hardness.

Findings

Exact solutions for up to 25 points achieved.

Proves the NP-hardness of both problems.

Highlights the computational challenge of area-optimal polygonizations.

Abstract

We consider methods for finding a simple polygon of minimum (Min-Area) or maximum (Max-Area) possible area for a given set of points in the plane. Both problems are known to be NP-hard; at the center of the recent CG Challenge, practical methods have received considerable attention. However, previous methods focused on heuristic methods, with no proof of optimality. We develop exact methods, based on a combination of geometry and integer programming. As a result, we are able to solve instances of up to n=25 points to provable optimality. While this extends the range of solvable instances by a considerable amount, it also illustrates the practical difficulty of both problem variants.