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.

翻译：我们研究了直切与方切披萨共享问题求解的计算复杂性。我们证明对于这两个问题，计算$\varepsilon$近似解是PPA完全的，而寻找方切问题的精确解是FIXP难的。我们的PPA难性结果适用于任意$\varepsilon < 1/5$的情况，即使所有权重分布均由非重叠的轴对齐矩形组成或为点集时亦然；我们的FIXP难性结果则适用于所有权重分布均为正方形与直角三角形的并集的情形。我们还证明了两个近似问题的判定变体均为NP完全的，而方切披萨共享精确版本的判定变体是$\exists\mathbb{R}$完全的。