COCOON 2016: Computing and Combinatorics pp 547-559

# Polygon Simplification by Minimizing Convex Corners

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9797)

## 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 $$\mathcal {A}_v(P'_{min}) = \mathcal {A}_v(P)$$.

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© Springer International Publishing Switzerland 2016

## Authors and Affiliations

• Yeganeh Bahoo
• 1
• Stephane Durocher
• 1
Email author
• J. Mark Keil
• 2
• Saeed Mehrabi
• 3
• Sahar Mehrpour
• 1
• Debajyoti Mondal
• 1
1. 1.Department of Computer ScienceUniversity of ManitobaWinnipegCanada