Pizza Sharing is PPA-hard

We study the computational complexity of finding a solution for the straight-cut and square-cut pizza sharing problems. We show that computing an ε-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 ε < 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.