On Simple Polygonalizations with Optimal Area

Abstract

We discuss the problem of finding a simple polygonalization for a given set of vertices P that has optimal area. We show that these problems are very closely related to problems of optimizing the number of points from a set Q in a simple polygon with vertex set P and prove that it is NP-complete to find a minimum weight polygon or a maximum weight polygon for a given vertex set, resulting in a proof of NP-completeness for the corresponding area optimization problems. This answers a generalization of a question stated by Suri in 1989. Finally, we turn to higher dimensions, where we prove that, for 1 ≤ k ≤ d, 2 ≤ d, it is NP-hard to determine the smallest possible total volume of the k-dimensional faces of a d-dimensional simple nondegenerate polyhedron with a given vertex set, answering a generalization of a question stated by O'Rourke in 1980.

On Polygons Enclosing Point Sets II

“…How tight is this bound? Inspired by the techniques in [9], we conclude this work observing that by using Pick's Theorem and allowing collinearities we can show that the mentioned bound, up to a constant, is tight. We state first Pick's theorem; recall that a point with integer coordinates is called a lattice point.…”

“…Fekete [9] considers the problem of finding polygonizations of point sets S that minimize or maximize the enclosed area. He proves that finding such polygonizations is NPcomplete.…”

On Optimal $w$-gons in Convex Polygons

“…Finding a possibly non-convex polygon of size w with minimum/maximum area is known as polygonalization, and is NP-complete even in R 2 [8]. We note that it is not always possible to find a strictly convex w-gon on any point set [9].…”

Greedy and Local Search Heuristics to Build Area-Optimal Polygons

“…Optimal area polygonalization resembles to the well-known travelling salesman problem, the difference being that the objective function of the former is the area of the computed polygon instead of its perimeter. This problem has been the subject of the 2019 Geometric Optimization Challenge and is known to be NP-hard for both minimization and maximization [2]. Exact algorithms are discussed in [3] and a recent state of the art is given in [1].…”