On the Reflexivity of Point Sets

TL;DR

This paper introduces a new measure called reflexivity for planar point sets, analyzing its properties, computational complexity, and algorithms for approximation, along with related convex covering and partitioning quantities.

Contribution

It defines reflexivity as a measure of how far a point set is from convexity, providing bounds, algorithms, and complexity results for computing it and related quantities.

Findings

Reflexivity is NP-complete to compute exactly.

Efficient algorithms exist for approximate computation of reflexivity.

The paper establishes bounds and approximation algorithms for convex cover and partition numbers.

Abstract

We introduce a new measure for planar point sets S that captures a combinatorial distance that S is from being a convex set: The reflexivity rho(S) of S is given by the smallest number of reflex vertices in a simple polygonalization of S. We prove various combinatorial bounds and provide efficient algorithms to compute reflexivity, both exactly (in special cases) and approximately (in general). Our study considers also some closely related quantities, such as the convex cover number kappa_c(S) of a planar point set, which is the smallest number of convex chains that cover S, and the convex partition number kappa_p(S), which is given by the smallest number of convex chains with pairwise-disjoint convex hulls that cover S. We have proved that it is NP-complete to determine the convex cover or the convex partition number and have given logarithmic-approximation algorithms for determining each.