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 ⊆ P′ each connected component R′ of P′ subsumes a distinct component R of P, i.e., R ⊆ 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 Ae(P) (resp., Av(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 Ae(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 Ae(P′min) = Ae(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 Ae(P′min) = Ae(P) and subsuming chains are of length three. Both results extend to the scenario when Av(P′min) = Av(P).