We investigate the problem of fairly dividing a divisible heterogeneous resource, also known as a cake, among a set of agents who may have different entitlements. We characterize the existence of a connected strongly-proportional allocation -- one in which every agent receives a contiguous piece worth strictly more than their proportional share. The characterization is supplemented with an algorithm that determines its existence using O(n * 2^n) queries. We devise a simpler characterization for agents with strictly positive valuations and with equal entitlements, and present an algorithm to determine the existence of such an allocation using O(n^2) queries. We provide matching lower bounds in the number of queries for both algorithms. When a connected strongly-proportional allocation exists, we show that it can also be computed using a similar number of queries. We also consider the problem of deciding the existence of a connected allocation of a cake in which each agent receives a piece worth a small fixed value more than their proportional share, and the problem of deciding the existence of a connected strongly-proportional allocation of a pie.

翻译：本文研究在代理人可能具有不同权益的情况下，公平分配一种可分割的异质资源（亦称"蛋糕"）的问题。我们刻画了连通强比例分配的存在性——在此类分配中，每个代理人均获得一个连续份额，其价值严格高于其比例份额。该刻画辅以一个算法，该算法使用 O(n * 2^n) 次查询即可判定其存在性。针对具有严格正估值且权益相等的代理人，我们提出了一种更简洁的刻画，并给出了一个使用 O(n^2) 次查询判定此类分配存在性的算法。我们为两种算法的查询次数提供了匹配的下界。当连通强比例分配存在时，我们证明其亦可通过类似数量的查询计算得出。我们还考虑了判定是否存在一种蛋糕的连通分配使得每个代理人获得份额价值超过其比例份额一个小的固定值的问题，以及判定是否存在饼的连通强比例分配的问题。