The existence of EFX allocations is a major open problem in fair division, even for additive valuations. The current state of the art is that no setting where EFX allocations are impossible is known, and EFX is known to exist for ($i$) agents with identical valuations, ($ii$) 2 agents, ($iii$) 3 agents with additive valuations, ($iv$) agents with one of two additive valuations and ($v$) agents with two-valued instances. It is also known that EFX exists if one can leave $n-1$ items unallocated, where $n$ is the number of agents. We develop new techniques that allow us to push the boundaries of the enigmatic EFX problem beyond these known results, and, arguably, to simplify proofs of earlier results. Our main results are ($i$) every setting with 4 additive agents admits an EFX allocation that leaves at most a single item unallocated, ($ii$) every setting with $n$ additive valuations has an EFX allocation with at most $n-2$ unallocated items. Moreover, all of our results extend beyond additive valuations to all nice cancelable valuations (a new class, including additive, unit-demand, budget-additive and multiplicative valuations, among others). Furthermore, using our new techniques, we show that previous results for additive valuations extend to nice cancelable valuations.

中文翻译：

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（几乎已满）存在四个代理的EFX（及以后）

EFX分配的存在是公平分配中的一个主要开放问题，即使对于附加估值也是如此。当前的现状是，没有已知的情况无法进行EFX分配，并且已知存在（E $）具有相同估值的（$ i $）代理商，（$ ii $）2个代理商，（$ iii $）3个代理商的EFX。具有加性估值的（$ iv $）代理具有两个加性估值之一，具有（$ v $）代理具有两个值实例。还众所周知，如果可以让$ n-1 $个项目不分配，则EFX存在，其中$ n $是代理的数量。我们开发了新的技术，这些技术使我们能够将神秘的EFX问题的范围推到这些已知结果之外，并且可以说是简化了早期结果的证明。我们的主要结果是（$ i $）每个设置有4种添加剂的设置都允许EFX分配，最多保留单个项目未分配，（$ ii $）每个具有$ n $附加估值的设置都有一个EFX分配，其中最多有$ n-2 $个未分配项目。此外，我们所有的结果都超出了加性估值，到所有可取消的良好估值（一个新类别，包括加性，单位需求，预算可加性和乘性估值等）。此外，使用我们的新技术，我们证明了以前的附加估值结果可扩展到不错的可取消估值。