# Locally EFX Allocations Over a Graph

The fairness notion of envy-free up to any good (EFX) has recently gained popularity in the fair allocation literature. However, few positive results about EFX allocations exist, with the existence of such allocations alone being an open question. In this work, we study an intuitive relaxation of EFX allocations: our allocations are only required to satisfy the EFX constraint for certain pre-defined pairs of agents. Since these problem instances can be represented using an undirected graph where the vertices correspond to the agents and each edge represents an EFX constraint, we call such allocations graph EFX (G-EFX) allocations. We show that G-EFX allocations exist for three different classes of graphs – two of them generalize the star K_1, n-1 and the third generalizes the three-edge path P_4. We also present and evaluate an algorithm using problem instances from Spliddit to show that G-EFX allocations are likely to exist for larger classes of graphs like long paths P_n.

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