Computing Envy-Free up to Any Good (EFX) Allocations via Local Search

We present a simple local search algorithm for computing EFX (envy-free up to any good) allocations of $m$ indivisible goods among $n$ agents with additive valuations. EFX is a compelling fairness notion, and whether such allocations always exist remains a major open question in fair division. Our algorithm employs simulated annealing with the total number of EFX violations as an objective function together with a single-transfer neighborhood structure to move through the space of allocations. It found an EFX allocation in all the instances tested, which included thousands of randomly generated inputs, and scaled to settings with hundreds of agents and/or thousands of items. The algorithm's simplicity, along with its strong empirical performance makes it a simple benchmark for evaluating future approaches. On the theoretical side, we provide a potential function for identical additive valuations, which ensures that any strict-descent procedure under the single-transfer neighborhood ends at an EFX allocation. This represents an alternative proof of existence for identical valuations.