We study the fair allocation of indivisible goods among agents, with a focus on limiting envy. A central fairness notion is envy-freeness up to any good (EFX), which requires that any envy toward another agent vanishes after the removal of any single good from the latter's bundle. The existence of EFX allocations is considered a major open problem in fair division. So far, it has only been established in limited settings. Christodoulou et al. [2023] proved the existence of EFX allocations for graphical valuations. In this setting, the agents correspond to the nodes of an underlying graph, and the goods correspond to the edges, and any good has positive value only for the endpoints of the corresponding edge. Their proof crucially relies on the restriction that the graph is simple, meaning that for any pair of agents, there is at most one good that has value to both. For multigraph valuations, where multiple goods may be valued by the same pair of agents, only partial results are known. Amanatidis et al. [2024] and Kaviani et al. [2025] obtained 2/3 and sqrt(2)/2 approximations of EFX, respectively; Kaviani et al. [2024] established existence under restricted additive valuations; and Afshinmehr et al. [2025a] proved existence under the assumption that the shortest cycle containing non-parallel edges has length at least 4. In this paper, we resolve this open problem by proving the existence of EFX allocations for multigraph instances under cancelable valuations, a strict superclass of additive valuation functions. Our proof is algorithmic and computes such allocations in polynomial time when the valuation functions are cancelable. This work contributes to the small number of EFX existence results that apply to an arbitrary number of agents.

翻译：暂无翻译