Pizza Sharing is PPA-hard

TL;DR

This paper investigates the computational complexity of pizza sharing problems, proving PPA-completeness for approximate solutions and FIXP-hardness for exact solutions, with implications for various geometric mass distributions.

Contribution

It establishes the complexity classifications of both approximate and exact pizza sharing problems, including PPA-completeness and FIXP-hardness results for specific geometric cases.

Findings

Computing an ε-approximate solution is PPA-complete for both problems.

Finding an exact solution for the square-cut problem is FIXP-hard.

Decision variants of approximate problems are NP-complete.

Abstract

We study the computational complexity of finding a solution for the straight-cut and square-cut pizza sharing problems. We show that computing an $\varepsilon$-approximate solution is PPA-complete for both problems, while finding an exact solution for the square-cut problem is FIXP-hard. Our PPA-hardness results apply for any $\varepsilon < 1/5$, even when all mass distributions consist of non-overlapping axis-aligned rectangles or when they are point sets, and our FIXP-hardness result applies even when all mass distributions are unions of squares and right-angled triangles. We also prove that the decision variants of both approximate problems are NP-complete, while the decision variant for the exact version of square-cut pizza sharing is $\exists\mathbb{R}$-complete.