Pushing the Frontier on Approximate EFX Allocations
This paper studies the existence of α-envy-free up to any good (α-EFX) allocations for indivisible goods under additive valuations. Addressing a long-standing approximation-ratio bottleneck—where the best prior guarantee was ≈0.618—we establish, for the first time, that a 2/3-EFX allocation always exists under three broad generalizations: (1) at most seven agents; (2) trivalued valuation functions (i.e., each agent’s values for goods take only three distinct values); and (3) valuations representable via multigraph structures. Methodologically, we integrate combinatorial game theory, discrete optimization, and constructive existence proofs to develop a novel analytical framework grounded in valuation-structure modeling. Our results not only break the previous approximation barrier but also systematically extend exact EFX existence theory—from restricted settings (e.g., two agents or binary valuations)—to more realistic multi-agent, multivalued, and structured valuation domains, thereby significantly broadening the applicability frontier of EFX fairness.