# Fair Division of Time: Multi-layered Cake Cutting

arXiv: https://arxiv.org/abs/2004.13397 · HTML: https://mlsanity.com/paper/2004.13397/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Apr 2020.

## Built on by (most cited first, 4 of 4)

- [Resource Allocation under the Latin Square Constraint](https://mlsanity.com/paper/2501.06506/) (2025)
- [Simultaneously Fair Allocation of Indivisible Items Across Multiple Dimensions](https://mlsanity.com/paper/2506.21727/) (2025)
- [Fair Division of Multi-layered Cakes](https://mlsanity.com/paper/2208.00726/) (2022)
- [The Complexity of Envy-Free Graph Cutting](https://mlsanity.com/paper/2312.07043/) (2023)

## Abstract (arXiv)

We initiate the study of multi-layered cake cutting with the goal of fairly allocating multiple divisible resources (layers of a cake) among a set of agents. The key requirement is that each agent can only utilize a single resource at each time interval. Several real-life applications exhibit such restrictions on overlapping pieces; for example, assigning time intervals over multiple facilities and resources or assigning shifts to medical professionals. We investigate the existence and computation of envy-free and proportional allocations. We show that envy-free allocations that are both feasible and contiguous are guaranteed to exist for up to three agents with two types of preferences, when the number of layers is two. We also show that envy-free feasible allocations where each agent receives a polynomially bounded number of intervals exist for any number of agents and layers under mild conditions on agents' preferences. We further devise an algorithm for computing proportional allocations for any number of agents and layers.