Erd\H{o}s-Szekeres and Testing Weak epsilon-Nets are NP-hard in 3 dimensions - and what now?

TL;DR

This paper proves that determining the largest convex subset and testing weak epsilon-nets in three dimensions are NP-hard and co-NP-hard respectively, highlighting computational complexity challenges beyond the planar case.

Contribution

It establishes NP-hardness for 3D Erd ext{H}os-Szekeres problems and co-NP-hardness for weak epsilon-net testing, answering longstanding open questions.

Findings

NP-hardness of largest convex subset in 3D

co-NP-hardness of weak epsilon-net testing in 3D

Contrast with polynomial-time solvability in 2D

Abstract

We consider the computational versions of the Erd\H os-Szekeres theorem and related problems in 3 dimensions. We show that, in constrast to the planar case, no polynomial time algorithm exists for determining the largest (empty) convex subset (unless P=NP) among a set of points, by proving that the corresponding decision problem is NP-hard. This answers a question by Dobkin, Edelsbrunner and Overmars from 1990. As a corollary, we derive a similar result for the closely related problem of testing weak epsilon-nets in R^3. Answering a question by Chazelle et al. from 1995, our reduction shows that the problem is co-NP-hard. This is work in progress - we are still trying to find a smart approximation algorithm for the problems.