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A Simpler Approach to the EFX Problem Envy-freeness up to any item (EFX) has emerged as a compelling fairness notion in discrete fair division. However, its existence remains one of the biggest open problems in the field. In a breakthrough, Chaudhury et al. (2020) establish the existence of EFX allocations for three agents with additive valuations through intricate case analysis. The paper “EFX: A Simpler Approach and an (Almost) Optimal Guarantee via Rainbow Cycle Number” by Akrami, Alon, Chaudhury, Garg, Mehlhorn, and Mehta offers a simpler approach for improving the EFX guarantee. They demonstrate the existence of EFX allocations for three agents when at least one has additive valuations (whereas the other two have general monotone valuations). Additionally, they nearly resolve a conjecture regarding the rainbow cycle number, leading to an (almost) tight bound for the existence of approximate EFX allocations with few unallocated items achievable through this approach. 

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 an EFX allocation has not been settled and is one of the most important problems in fair division. Toward resolving this question, many impressive results show the existence of its relaxations. In particular, it is known that 0.618-EFX allocations exist and that EFX allocation exists if we do not allocate at most (n-1) goods. Reducing the number of unallocated goods has emerged as a systematic way to tackle the main question. For example, follow-up works on three- and four-agents cases, respectively, allocated two more unallocated goods through an involved procedure. In this paper, we study the general case and achieve sublinear numbers of unallocated goods. Through a new approach, we show that for every [Formula: see text], there always exists a [Formula: see text]-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 define the notion of rainbow cycle number [Formula: see text] in directed graphs. For all [Formula: see text] is the largest k such that there exists a k-partite graph [Formula: see text], in which each part has at most d vertices (i.e., [Formula: see text] for all [Formula: see text]); for any two parts V_iand V_j, each vertex in V_ihas an incoming edge from some vertex in V_jand vice versa; and there exists no cycle in G that contains at most one vertex from each part. We show that any upper bound on [Formula: see text] directly translates to a sublinear bound on the number of unallocated goods. We establish a polynomial upper bound on [Formula: see text], yielding our main result. Furthermore, our approach is constructive, which also gives a polynomial-time algorithm for finding such an allocation. Funding: J. Garg was supported by the Directorate for Computer and Information Science and Engineering [Grant CCF-1942321]. R. Mehta was supported by the Directorate for Computer and Information Science and Engineering [Grant CCF-1750436].