A Complete Landscape of EFX Allocations on Graphs: Goods, Chores and Mixed Manna

We study envy-free up to any item (EFX) allocations on graphs where vertices and edges represent agents and items respectively. An agent is only interested in items that are incident to her and all other items have zero marginal values to her. Christodoulou et al. first proposed this setting and studied the case of goods. We extend this setting to the case of mixed manna where an item may be liked or disliked by its endpoint agents. In our problem, an agent has an arbitrary valuation over her incident items such that the items she likes have non-negative marginal values to her and those she dislikes have non-positive marginal values. We provide a complete study of the four notions of EFX for mixed manna in the literature, which differ by whether the removed item can have zero marginal value. We prove that an allocation that satisfies the notion of EFX where the virtually-removed item could always have zero marginal value may not exist and determining its existence is NP-complete, while one that satisfies any of the other three notions always exists and can be computed in polynomial time. We also prove that an orientation (i.e., a special allocation where each edge must be allocated to one of its endpoint agents) that satisfies any of the four notions may not exist, and determining its existence is NP-complete.