Pizza Sharing Is PPA-Hard

Abstract

We study the computational complexity of computing solutions for the straight-cut and square-cut pizza sharing problems. We show that finding an approximate solution is PPA-hard for the straight-cut problem, and PPA-complete for the square-cut problem, while finding an exact solution for the square-cut problem is FIXP-hard and in BU. Our PPA-hardness results apply even when all mass distributions are unions of non-overlapping squares, and our FIXP-hardness result applies even when all mass distributions are unions of weighted squares and right-angled triangles. We also prove that decision variants of the square-cut problem are hard: the approximate problem is NP-complete, and the exact problem is ETR-complete.

Constant inapproximability for PPA

“…Both problems were proven to be PPA-complete when ε is inversely polynomial and PPAD-hard for a small constant ε ∈ (0, 1) via direct reductions from Consensus-Halving [Deligkas et al, 2020;Schnider, 2021]. Using the reductions from [Deligkas et al, 2020], which increase the error by at most a polynomially small amount, alongside our main theorem yields the following.…”

“…This was the first "natural" complete problem for the class and it has been pivotal in proving further such completeness results. For example, PPA-completeness has since been shown for other "natural" problems such as the necklace splitting problem and the discrete ham sandwich problem [Filos-Ratsikas and Goldberg, 2019], two types of the pizza-sharing problem [Deligkas et al, 2020;Schnider, 2021], and finding fair independent sets in cycles and paths [Haviv, 2021]. We refer to these as natural problems since their definition does not involve any kind of circuit, as opposed to "unnatural" problems like Tucker (the problem associated with Tucker's Lemma), which was already known to be PPA-complete [Aisenberg et al, 2020], but whose definition involves a Boolean circuit.…”

A Practical Algorithm with Performance Guarantees for the Art Gallery Problem

“…The complexity class ∃R is important as it gives a precise characterization of many important algorithmic problems. Important algorithmic problems are related to graph drawing [59,18,54,27,34,26,60], the art gallery problem [4,75], geometric packing [7], linkages [1], polytopes [66,36], machine learning [5,16,80], matrix factorization [70,72,73], order types [74,63] and various other topics [?, 42,49,68,69,33,6,40,48,62,47]. None of these algorithmic problems are known to be contained in NP.…”

Multi-agent systems for computational economics and finance

“…In [DFHM22a] we showed that the problem is PPA-complete even for computing a relaxed solution where the agents believe that the di erence in value of the two parts is at most a large constant value. Our work has important complexity implications for other major problems in fair division of divisible and indivisible resources, namely discrete ham-sandwich, necklace spli ing [FRG22], pizza sharing [DFM22], and more. Prior to this, in [DFMS21] we studied the problem of nding an exact solution of consensus halving and showed that its complexity di ers tremendously from its approximate version.…”