We consider the fair division problem of indivisible chores and resolve the long-standing open problem for the existence of EFX allocations with additive cost functions. We show that, even for tri-valued additive cost functions, for every $n\geq 4$, there exists an instance with $n$ agents where no EFX allocation exists. Our counterexample only uses three types of chores, which is also tight on the number of types, as an EFX allocation is known to exist for two types of chores. We then consider bi-valued instances. We show that, for every $n\geq 4$, there exists an instance with $n$ agents where every EFX allocation is not Pareto-optimal. This is also the first example showing the incompatibility of EFX and Pareto-optimality when the costs of items are positive: existing examples showing the incompatibility of EFX and Pareto-optimal exploit items with $0$ costs. Our result shows such an example exists even for bi-valued instances. The number of agents $n$ is also tight: for $n\leq 3$, it is known that EFX is compatible with Pareto-optimality. Finally, we also show that an EFX allocation is guaranteed to exist for $n=4$.

翻译：我们考虑不可分割家务的公平分配问题，并解决了加性成本函数下EFX分配存在性的长期未解难题。研究表明，即使对于三值加性成本函数，对于每个 $n\geq 4$，都存在一个包含 $n$ 个智能体的实例，其中不存在EFX分配。我们的反例仅使用三种类型的家务，这在类型数量上也是紧的，因为已知对于两种类型的家务存在EFX分配。接着我们考虑双值实例。我们证明，对于每个 $n\geq 4$，都存在一个包含 $n$ 个智能体的实例，其中每个EFX分配都不是帕累托最优的。这也是首个表明当物品成本为正时EFX与帕累托最优性不兼容的例子：现有证明EFX与帕累托最优不兼容的例子利用了成本为 $0$ 的物品。我们的结果表明，即使在双值实例中也存在这样的例子。智能体数量 $n$ 也是紧的：对于 $n\leq 3$，已知EFX与帕累托最优性是兼容的。最后，我们还证明对于 $n=4$，EFX分配的存在性是有保证的。