AI 中文总结
本文提出存在性转移框架，将连续公平分配定理转化为不可分物品分配的EFk类保证，结合相关定理得到多种公平分配结果，包括非加性估值下的共识公平分配的EFk类保证等。
AI 中文摘要
我们提出一种存在性转移框架，用于将连续公平分配定理转化为沿路径排列的不可分物品的分配保证。该框架可使连续无嫉妒（envy-freeness）和共识结果直接转化为不可分物品分配的EFk类保证。将此方法与连通切蛋糕定理结合，针对相同估值下的主体，以及主体数量为素数幂时的任意估值，我们得到满足“最多一件物品的无嫉妒”（envy-freeness up to one good）和“最多一项任务的无嫉妒”（envy-freeness up to one chore）的连通分配。将此方法与Jojić等人的等基数项链分割定理结合，我们证明：对于任意素数幂个束r和n个任意估值函数，存在一种分配方式，其中每个束是最多n个区间的并集，且这些束满足“最多n件物品的共识”（consensus up to n goods）和“最多n项任务的共识”（consensus up to n chores）。该结果是除平分情况外，首个针对非加性估值的共识公平分配的EFk类保证。若同时施加无嫉妒约束，每个保证需多增加一个区间和一件物品；作为推论，当主体数量为素数幂时，所有具有单调估值的实例均存在一种EF2分配，且各束大小差异不超过2。
英文摘要
We give an existential transfer framework for converting continuous fair division theorems into guarantees for indivisible items arranged on a path. This allows continuous envy-freeness and consensus results to translate directly into EF$k$-type guarantees for indivisible allocations. Combining this method with connected cake-cutting theorems, we obtain connected allocations satisfying envy-freeness up to one good and one chore for identical valuations and for arbitrary valuations when the number of agents is a prime power. Combining this method with the equicardinal necklace-splitting theorem of Jojić et al., we show that, for any prime-power number $r$ of bundles and $n$ arbitrary valuation functions, there exists an allocation in which every bundle is the union of at most $n$ intervals, and the bundles satisfy consensus up to $n$ goods and $n$ chores. This result is the first EF$k$-type guarantee for consensus fair division with non-additive valuations beyond the halving case. Envy-freeness constraints can be imposed simultaneously at the cost of one additional interval and one additional item in each guarantee. As a consequence, when the number of agents is a prime power, every instance with monotone valuations admits an EF$2$ allocation whose bundle sizes differ by at most two.
