The complexity of sharing a pizza

TL;DR

This paper investigates the computational complexity of fairly sharing a 2D pizza with multiple ingredients through straight cuts, proving the problem is PPA-complete and providing new proofs and hardness results.

Contribution

It establishes the PPA-completeness of the problem when ingredients are point sets and offers a new non-topological proof that n cuts suffice, along with hardness results and higher-dimensional variants.

Findings

The problem is PPA-complete for point set ingredients.

A new proof shows n cuts suffice without topological methods.

Hardness results and higher-dimensional variants are also established.

Abstract

Assume you have a 2-dimensional pizza with $2n$ ingredients that you want to share with your friend. For this you are allowed to cut the pizza using several straight cuts, and then give every second piece to your friend. You want to do this fairly, that is, your friend and you should each get exactly half of each ingredient. How many cuts do you need? It was recently shown using topological methods that $n$ cuts always suffice. In this work, we study the computational complexity of finding such $n$ cuts. Our main result is that this problem is PPA-complete when the ingredients are represented as point sets. For this, we give a new proof that for point sets $n$ cuts suffice, which does not use any topological methods. We further prove several hardness results as well as a higher-dimensional variant for the case where the ingredients are well-separated.