Two-sided convexity testing with certificates

TL;DR

This paper improves a randomized property testing algorithm for convexity in point sets, providing certificates and extending its applicability to higher dimensions with sublinear query complexity.

Contribution

The authors redesign and analyze a convexity testing algorithm, adding certificates and extending its input range for higher dimensions, while maintaining sublinear query complexity.

Findings

Accepts convex sets with high probability

Rejects non-convex sets with high probability and provides a witness

Outputs an approximation of the largest convex subset

Abstract

We revisit the problem of property testing for convex position for point sets in $\mathbb{R}^d$. Our results draw from previous ideas of Czumaj, Sohler, and Ziegler (ESA 2000). First, the algorithm is redesigned and its analysis is revised for correctness. Second, its functionality is expanded by (i)~exhibiting both negative and positive certificates along with the convexity determination, and (ii)~significantly extending the input range for moderate and higher dimensions. The behavior of the randomized tester is as follows: (i)~if $P$ is in convex position, it accepts; (ii)~if $P$ is far from convex position, with probability at least $2/3$, it rejects and outputs a $(d+2)$-point witness of non-convexity as a negative certificate; (iiii)~if $P$ is close to convex position, with probability at least $2/3$, it accepts and outputs an approximation of the largest subset in convex position. The algorithm examines a sublinear number of points and runs in subquadratic time for every dimension $d$.