Communication Complexity of Cake Cutting

TL;DR

This paper investigates the communication complexity of discrete cake-cutting problems, classifying them into easy, medium, and hard categories, and proves certain problems are not in the easiest class, highlighting open challenges.

Contribution

It introduces a discrete model for cake-cutting communication complexity and classifies problems into complexity classes, providing new lower bounds for specific fair division problems.

Findings

Perfect allocation for 2 players is not in the easy class.

Equitable allocation for any number of players is not in the easy class.

Proposes a framework for classifying cake-cutting problems by communication complexity.

Abstract

We study classic cake-cutting problems, but in discrete models rather than using infinite-precision real values, specifically, focusing on their communication complexity. Using general discrete simulations of classical infinite-precision protocols (Robertson-Webb and moving-knife), we roughly partition the various fair-allocation problems into 3 classes: "easy" (constant number of rounds of logarithmic many bits), "medium" (poly-logarithmic total communication), and "hard". Our main technical result concerns two of the "medium" problems (perfect allocation for 2 players and equitable allocation for any number of players) which we prove are not in the "easy" class. Our main open problem is to separate the "hard" from the "medium" classes.