# Polygon Simplification by Minimizing Convex Corners

## TL;DR

This paper investigates the complexity of simplifying polygons by minimizing convex corners, proving NP-completeness for polygons with holes and providing polynomial solutions for simple polygons under certain restrictions.

## Contribution

It establishes the NP-hardness of finding minimal convex corners in polygons with holes and identifies conditions under which the problem becomes polynomial-time solvable for simple polygons.

## Key findings

- NP-complete for polygons with holes
- Polynomial-time solvable for simple polygons with restrictions
- Complexity remains open for general simple polygons

## Abstract

Let $P$ be a polygon with $r>0$ reflex vertices and possibly with holes and islands. A subsuming polygon of $P$ is a polygon $P'$ such that $P \subseteq P'$, each connected component $R$ of $P$ is a subset of a distinct connected component $R'$ of $P'$, and the reflex corners of $R$ coincide with those of $R'$. A subsuming chain of $P'$ is a minimal path on the boundary of $P'$ whose two end edges coincide with two edges of $P$. Aichholzer et al. proved that every polygon $P$ has a subsuming polygon with $O(r)$ vertices, and posed an open problem to determine the computational complexity of computing subsuming polygons with the minimum number of convex vertices. We prove that the problem of computing an optimal subsuming polygon is NP-complete, but the complexity remains open for simple polygons (i.e., polygons without holes). Our NP-hardness result holds even when the subsuming chains are restricted to have constant length and lie on the arrangement of lines determined by the edges of the input polygon. We show that this restriction makes the problem polynomial-time solvable for simple polygons.