Feige & Fine solve fair allocation for submodular valuations with EF1+MMS

In a new arXiv paper, computer scientists Uriel Feige and Assaf Fine tackle a fundamental problem in fair division: allocating indivisible items to agents with equal claims. Two dominant fairness notions exist: share-based (maximin share, or MMS) which guarantees each agent a certain fraction of what they could get by partitioning items, and comparison-based (envy-freeness, or EF1) which ensures no agent envies another after removing one item. Historically, these notions conflict—optimal MMS allocations often violate EF1, and vice versa. The breakthrough: Feige and Fine show that for submodular valuations, a class that captures diminishing returns and is strictly more general than additive valuations, one can simultaneously achieve EF1 (and even the stronger EFL) with a constant-factor MMS guarantee. This is the first time such simultaneous guarantees have been proven beyond additive valuations.

The authors use a novel combinatorial algorithm that leverages the structure of submodular functions, combining greedy assignment with careful rebalancing. The result holds for any number of agents and items, with the MMS approximation factor being a constant independent of the number of agents. This opens up practical applications in resource allocation (e.g., cloud computing, course assignment, or data partitioning) where agent preferences exhibit diminishing returns. The work bridges a gap between theoretical computer science and real-world mechanisms, showing that strong fairness is achievable even when valuations are complex.