Polygon Simplification by Minimizing Convex Corners

Abstract

Let P be a polygon with $r>0$ reflex vertices and possibly with holes. A subsuming polygon of P is a polygon $P'$ such that $P \subseteq P'$, each connected component $R'$ of $P'$ subsumes a distinct component R of P, i.e., $R\subseteq R'$, and the reflex corners of R coincide with the reflex corners 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. Let $\mathcal {A}_e(P)$ (resp., $\mathcal {A}_v(P)$) be the arrangement of lines determined by the edges (resp., pairs of vertices) of P. Aichholzer et al. observed that a challenge of computing an optimal subsuming polygon $P'_{min}$, i.e., a subsuming polygon with minimum number of convex vertices, is that it may not always lie on $\mathcal {A}_e(P)$. We prove that in some settings, one can find an optimal subsuming polygon for a given simple polygon in polynomial time, i.e., when $\mathcal {A}_e(P'_{min}) = \mathcal {A}_e(P)$ and the subsuming chains are of constant length. In contrast, we prove the problem to be NP-hard for polygons with holes, even if there exists some $P'_{min}$ with $\mathcal {A}_e(P'_{min}) = \mathcal {A}_e(P)$ and subsuming chains are of length three. Both results extend to the scenario when $\mathcal {A}_v(P'_{min}) = \mathcal {A}_v(P)$.

S. Durocher—Work of the author is supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).