Decremental Greedy Polygons and Polyhedra Without Sharp Angles

TL;DR

This paper presents efficient algorithms for finding max-min-angle polygons and polyhedra in point sets, explores the complexity of related problems, and generalizes decremental greedy algorithms for optimal solutions.

Contribution

It introduces new algorithms for max-min-angle polygons and polyhedra, analyzes their complexity, and formalizes a class of problems solvable by decremental greedy methods.

Findings

Max-min-angle polygon in planar sets can be found in O(n log n) time.

Max-min-solid-angle convex polyhedron in 3D can be found in O(n^2) time.

Finding the max-min-angle polygonal curve in 3D is NP-hard without repetitions.

Abstract

We show that the max-min-angle polygon in a planar point set can be found in time $O(n\log n)$ and a max-min-solid-angle convex polyhedron in a three-dimensional point set can be found in time $O(n^2)$. We also study the maxmin-angle polygonal curve in 3d, which we show to be $\mathsf{NP}$-hard to find if repetitions are forbidden but can be found in near-cubic time if repeated vertices or line segments are allowed, by reducing the problem to finding a bottleneck cycle in a graph. We formalize a class of problems on which a decremental greedy algorithm can be guaranteed to find an optimal solution, generalizing our max-min-angle and bottleneck cycle algorithms, together with a known algorithm for graph degeneracy.