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

2026-05-20

AI Summary

This work investigates the minimum query complexity required by randomized algorithms to achieve proportional fair cake cutting under the Robertson–Webb query model. By integrating information-theoretic arguments with adversarial techniques, the paper establishes the first rigorous lower bound on the query complexity for this problem. The main contribution is a proof that any randomized algorithm guaranteeing proportionality must perform at least $\Omega(n \log n)$ queries, thereby revealing a fundamental computational limitation inherent to proportional cake-cutting protocols. This result underscores the intrinsic difficulty of achieving fairness in resource allocation even when randomization is permitted, and it provides a theoretical foundation for understanding the trade-offs between fairness guarantees and computational efficiency in division problems.

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 $Ω(n \log n)$ queries.

Problem

• cake cutting
• proportional fairness
• randomized algorithm
• query complexity
• lower bound

Innovation

• cake cutting
• proportional fairness
• randomized lower bound
• query complexity
• Robertson-Webb model