Envy-Freeness up to Any Good Allocations on Graphs

We study envy-freeness up to any good (EFX) in settings where valuations can be represented via a graph of arbitrary size where vertices correspond to agents and edges to items. An item (edge) has zero marginal value to all agents (vertices) not incident to the edge. Each vertex may have an arbitrary monotone valuation on the set of incident edges. We first consider allocations that correspond to orientations of the edges, where we show that EFX does not always exist, and furthermore, that it is NP-complete to decide whether an EFX orientation exists. Our main result is that EFX allocations exist for this setting. This is one of the few cases where EFX allocations are known to exist for more than three agents. Funding: The research project is implemented in the framework of the Hellenic Foundation for Research and Innovation (HFRI) call “Basic research Financing (Horizontal Support of all Sciences)” under the National Recovery and Resilience Plan “Greece 2.0” funded by the European Union—NextGenerationEU HFRI Project 15635.