Maximum Weight Convex Polytope

TL;DR

This paper investigates the computational complexity of the maximum weight convex polytope problem, establishing NP-hardness in higher dimensions and providing a new algorithm for the 2D case.

Contribution

It proves NP-hardness for the problem in 3D and higher, and introduces a new algorithm for the 2D case matching previous complexity.

Findings

NP-hard in 3D and higher dimensions

Hard to approximate within n^{1/2-ε} in 4D+

New algorithm for 2D with O(n^3) complexity

Abstract

We study the maximum weight convex polytope problem, in which the goal is to find a convex polytope maximizing the total weight of enclosed points. Prior to this work, the only known result for this problem was an $O(n^3)$ algorithm for the case of $2$ dimensions due to Bautista et al. We show that the problem becomes $\mathcal{NP}$-hard to solve exactly in $3$ dimensions, and $\mathcal{NP}$-hard to approximate within $n^{1/2-\epsilon}$ for any $\epsilon > 0$ in $4$ or more dimensions. %\polyAPX-complete in $4$ dimensions even with binary weights. We also give a new algorithm for $2$ dimensions, albeit with the same $O(n^3)$ running time complexity as that of the algorithm of Bautsita et al.