We study the problem of dividing a multi-layered cake among heterogeneous agents under non-overlapping constraints. 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 $1/{\varepsilon}$.

翻译：我们研究在不重叠的限制下将一个多层蛋糕分解给不同代理人的问题。这个问题最近由Hosseini等人(2020年)提出,反映了若干自然情况,例如每个代理人在一段时间内分配多个设施,每个代理人可以同时在多数一个设施同时使用,以及分配任务,每个代理人可以同时执行多数一项任务。我们确定存在一个没有嫉妒的多层蛋糕,这种多层蛋糕是非重叠的,每个层蛋糕内毗连的,当代理人的美元是一个主要权力,而每层的美元数目最多为美元,从而对最近的开放问题提供积极的部分答案。为了实现这一目标,我们采用了基于一般固定点的新方法,最初由Volovikov(1996年)所证明,最近由Joji\'{c}、Panina和\v ⁇ ivaljevi\{c} (202020年)适用于一个蛋糕的无嫉妒分解问题。我们进一步表明,对于三个具有单一偏好偏好点的代理人之间的两层蛋糕分,因此,一个正部分部分对最近一个开放问题的答案是1美元和连续1美元。