Mind the Gap: Cake Cutting With Separation

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. Task Cake Cutting Pie Cutting Decide whether MMS i = r Yes (Cor. 3.9) No (Thm. 4.2) Decide whether MMS i > r Yes (Thm. 3.8) No (Thm. 4.2) Decide whether MMS i ≥ r Yes (Thm. 3.5) No (Thm. 4.5) Compute the maximin share of an agent No (Thm. 3.2) No (Cor. 4.9) Approximate the maximin share up to ε Yes (Cor. 3.6) Yes (Thm. 4.10) Compute a maximin partition of an agent No (Cor. 3.4) No (Cor. 4.9) Approximate a maximin partition up to ε Yes (Cor. 3.6) Yes (Thm. 4.10