An $\Omega(n \log n)$ Randomized Lower Bound for Cutting a Cake into Proportionally Fair Pieces

TL;DR

This paper establishes a fundamental lower bound of Omega(n log n) queries for any randomized algorithm solving the proportional cake cutting problem in the Robertson-Webb model.

Contribution

It proves a new theoretical lower bound on the query complexity for proportional cake cutting, advancing understanding of the problem's computational limits.

Findings

Any randomized cake cutting algorithm requires at least Omega(n log n) queries.

The lower bound applies specifically to the Robertson-Webb model.

Proportional fairness in cake cutting has inherent complexity constraints.

Abstract

We consider the classic cake cutting problem in the Robertson-Webb model, with the objective of proportional fairness. We show that any randomized algorithm must use $\Omega(n \log n)$ queries.