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We consider the problem of sharing a set of indivisible goods among agents ina fair manner, namely such that the allocation is envy-free up to any good(EFX). We focus on the problem of computing an EFX allocation in the two-agentcase and characterize the computational complexity of the problem for mostwell-known valuation classes. We present a simple greedy algorithm that solvesthe problem when the agent valuations are weakly well-layered, a class whichcontains gross substitutes and budget-additive valuations. For the next largestvaluation class we prove a negative result: the problem is PLS-complete forsubmodular valuations. All of our results also hold for the setting where thereare many agents with identical valuations.

The Frontier of Intractability for EFX with Two Agents