Researchers crack AI envy resolution with new algorithm
EEAG problem solved in polynomial time with one item type—NP-complete with two.
A new study establishes a sharp dichotomy for envy elimination by adding goods when the additional pool has bounded supply. With one additional item type, the problem is solvable in polynomial time for any number of agents; with exactly two item types, it becomes NP-complete, even when both types have positive finite supply and the approvers of one type form a subset of the approvers of the other. The paper also proves weak NP-completeness for just two agents with identical additive valuations, one initially endowed good, and a growing number of unit-supply item types. Both results close previously open cases in fair division.
- EEAG is polynomial-time solvable with one item type using Bellman-Ford over difference constraints
- NP-complete when two item types are added, even with finite supply and subset-structured approvals
- Weak NP-completeness proven for two agents with identical valuations and many item types
Why It Matters
Impacts fair division in AI, resource allocation, and multi-agent systems where envy-free allocations are critical for equitable outcomes.