Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

TL;DR

This paper explores the computational complexity of fair, contiguous cake cutting, presenting efficient algorithms for low-envy allocations and establishing NP-hardness for certain decision problems, with implications for both continuous and discrete settings.

Contribution

It introduces new algorithms for approximate envy-free allocations and proves NP-hardness results for various constrained cake cutting problems.

Findings

Efficient algorithms for low-envy contiguous cake allocations.

NP-hardness results for fixed agent orderings and cut constraints.

Hardness extensions to discretized, indivisible item settings.

Abstract

We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-free algorithm exists in this setting, we exhibit efficient algorithms that produce allocations with low envy among the agents. We then establish NP-hardness results for various decision problems on the existence of envy-free allocations, such as when we fix the ordering of the agents or constrain the positions of certain cuts. In addition, we consider a discretized setting where indivisible items lie on a line and show a number of hardness results extending and strengthening those from prior work. Finally, we investigate connections between approximate and exact envy-freeness, as well as between continuous and discrete cake cutting.