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# The Complexity of Sharing a Pizza

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LIPIcs.ISAAC.2021.13.pdf
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## Cite As

Patrick Schnider. The Complexity of Sharing a Pizza. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ISAAC.2021.13

## 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
##### Keywords
• pizza sharing
• Ham-Sandwich theorem
• PPA
• computational geometry
• computational complexity

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