This problem, recently proposed by Hosseini et al. (2020), captures several natural scenarios such as the allocation of multiple facilities over time where each agent can utilize at most one facility simultaneously, and the allocation of tasks over time where each agent can perform at most one task simultaneously. We establish the existence of an envy-free multi-division that is both non-overlapping and contiguous within each layered cake when the number $n$ of agents is a prime power and the number $m$ of layers is at most $n$, thus providing a positive partial answer to a recent open question. To achieve this, we employ a new approach based on a general fixed point theorem, originally proven by Volovikov (1996), and recently applied by Joji\'{c}, Panina, and \v{Z}ivaljevi\'{c} (2020) to the envy-free division problem of a cake. We further show that for a two-layered cake division among three agents with monotone preferences, an $\varepsilon$-approximate envy-free solution that is both non-overlapping and contiguous can be computed in logarithmic time of $1/{\varepsilon}$.

中文翻译：

---

多层次蛋糕的无忌分工

这个问题最近由 Hosseini 等人提出。(2020)，捕获了几个自然场景，例如随着时间的推移分配多个设施，其中每个代理最多可以同时使用一个设施，以及随着时间的推移分配任务，其中每个代理最多可以同时执行一项任务。当代理的数量 $n$ 是素数且层的数量 $m$ 至多为 $n$ 时，我们建立了一个无嫉妒的多分部的存在，该分部在每个分层蛋糕中既不重叠又连续，从而为最近的一个悬而未决的问题提供了肯定的部分答案。为了实现这一点，我们采用了一种基于一般不动点定理的新方法，最初由 Volovikov (1996) 证明，最近由 Joji\'{c}、Panina 和 \v{Z}ivaljevi\'{c} 应用(2020) 解决蛋糕的无嫉妒分割问题。