Improving Envy Freeness up to Any Good Guarantees Through Rainbow Cycle Number

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 epsilon is an element of(0, 1/2], there always exists a (1 - epsilon)-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 R(center dot) in directed graphs. For all d is an element of N, R(d) is the largest k such that there exists a k-partite graph G = (boolean OR V-i is an element of[k](i), E), in which each part has at most d vertices (i.e., |V-i| <= d for all i is an element of[k]); 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 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.