Few Cuts Meet Many Point Sets

TL;DR

This paper addresses partitioning multiple point sets in high-dimensional space with minimal shared hyperplanes, offering a logarithmic approximation via a greedy submodular optimization approach.

Contribution

It introduces a novel approximation method for partitioning point sets using shared hyperplanes, extending classical geometric theorems.

Findings

Logarithmic approximation achieved for the partitioning problem

Greedy algorithm effectively approximates optimal solutions

Connections established to the Ham-Sandwich Theorem

Abstract

We study the problem of how to breakup many point sets in $\mathbb{R}^d$ into smaller parts using a few splitting (shared) hyperplanes. This problem is related to the classical Ham-Sandwich Theorem. We provide a logarithmic approximation to the optimal solution using the greedy algorithm for submodular optimization.