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.

中文翻译：

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连续切蛋糕：硬度结果和近似算法

我们研究了蛋糕的公平分配，它作为可分割资源的隐喻，要求每个代理都应该收到一块连续的蛋糕。虽然已知在此设置中不存在有限无嫉妒算法，但我们展示了有效的算法，可以在代理之间产生具有低嫉妒的分配。然后，我们针对存在无嫉妒分配的各种决策问题建立 NP 硬度结果，例如当我们固定代理的顺序或限制某些切割的位置时。此外，我们考虑了一个离散化的设置，其中不可分割的项目位于一条线上，并显示了许多硬度结果，扩展和加强了先前工作中的硬度结果。最后，我们研究了近似和精确无嫉妒之间的联系，以及连续和离散蛋糕切割之间的联系。