We study the problem of fairly allocating a set of indivisible goods among $n$ agents with additive valuations. Envy-freeness up to any good (EFX) is arguably the most compelling fairness notion in this context. However, the existence of EFX allocations has not been settled and is one of the most important problems in fair division. Towards resolving this problem, many impressive results show the existence of its relaxations, e.g., the existence of $0.618$-EFX allocations, and the existence of EFX at most $n-1$ unallocated goods. The latter result was recently improved for three agents, in which the two unallocated goods are allocated through an involved procedure. Reducing the number of unallocated goods for arbitrary number of agents is a systematic way to settle the big question. In this paper, we develop a new approach, and show that for every $\varepsilon \in (0,1/2]$, there always exists a $(1-\varepsilon)$-EFX allocation with sublinear number of unallocated goods and high Nash welfare. For this, we reduce the EFX problem to a novel problem in extremal graph theory. We introduce the notion of rainbow cycle number $R(\cdot)$. For all $d \in \mathbb{N}$, $R(d)$ is the largest $k$ such that there exists a $k$-partite digraph $G =(\cup_{i \in [k]} V_i, E)$, in which 1) each part has at most $d$ vertices, i.e., $\lvert V_i \rvert \leq d$ for all $i \in [k]$, 2) for any two parts $V_i$ and $V_j$, each vertex in $V_i$ has an incoming edge from some vertex in $V_j$ and vice-versa, and 3) there exists no cycle in $G$ that contains at most one vertex from each part. We show that any upper bound on $R(d)$ directly translates to a sublinear bound on the number of unallocated goods. We establish a polynomial upper bound on $R(d)$, yielding our main result. Furthermore, our approach is constructive, which also gives a polynomial-time algorithm for finding such an allocation.

中文翻译：

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通过Rainbow Cycle Number改善EFX保证

我们研究了在具有加法估值的$ n $代理商之间公平分配一组不可分割商品的问题。在这种情况下，最令人羡慕的至善至美（EFX）无疑是最引人注目的公平概念。但是，EFX分配的存在尚未解决，这是公平分配中最重要的问题之一。为了解决这个问题，许多令人印象深刻的结果表明存在放宽的情况，例如存在$ 0.618 $ -EFX分配，并且存在最多$ n-1 $未分配商品的EFX。最近针对三个代理改进了后者的结果，其中两个未分配的货物通过涉及的程序进行分配。为任意数量的代理商减少未分配商品的数量是解决这个大问题的系统方法。在本文中，我们开发了一种新方法，我们表明，$ R（d）$的任何上限直接转换为未分配商品数量的亚线性界限。我们建立$ R（d）$的多项式上限，得出我们的主要结果。此外，我们的方法是建设性的，它还提供了多项式时间算法来查找这种分配。