We study the problem of fairly allocating a divisible resource, also known as cake cutting, with an additional requirement that the shares that different agents receive should be sufficiently separated from one another. This captures, for example, constraints arising from social distancing guidelines. While it is sometimes impossible to allocate a proportional share to every agent under the separation requirement, we show that the well-known criterion of maximin share fairness can always be attained. We then provide algorithmic analysis of maximin share fairness in this setting—for instance, the maximin share of an agent cannot be computed exactly by any finite algorithm, but can be approximated with an arbitrarily small error. In addition, we consider the division of a pie (i.e., a circular cake) and show that an ordinal relaxation of maximin share fairness can be achieved. We also prove that an envy-free or equitable allocation that allocates the maximum amount of resource exists under separation.

我们研究公平分配可分割资源的问题，也称为蛋糕切割，另外要求不同代理收到的份额应该彼此充分分离。例如，这捕获了由社交距离准则引起的限制。虽然有时不可能根据分离要求为每个代理分配比例份额，但我们表明，众所周知的最大份额公平标准总是可以实现的。然后，我们在此设置中提供了最大最小份额公平性的算法分析——例如，代理的最大份额不能通过任何有限算法精确计算，但可以用任意小的误差来近似。此外，我们考虑一个饼的划分（即，一个圆形蛋糕）并表明可以实现最大最小份额公平的序数松弛。我们还证明了在分离下存在分配最大资源量的无嫉妒或公平分配。