Polygon Simplification by Minimizing Convex Corners

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 ofR coincide with the reflex corners ofR′. 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 ).