Logarithmic Comparison-Query Complexity for EF1 Allocations

Abstract

We study the comparison-query complexity of computing allocations that are envy-free up to one item (EF1) for $n$ agents with heterogeneous additive preferences over $m$ indivisible items. In this model, the algorithm may only query which of two bundles an agent prefers, without ever observing numerical valuations or costs. Prior work established $O(\log m)$ query bounds for at most three agents, leaving open whether query complexity logarithmic in $m$ is achievable for every fixed $n \ge 4$. We resolve this open question in the affirmative: for every fixed $n\ge2$, we design algorithms that compute EF1 allocations using $O(\log m)$ comparison queries, matching the known $Ω(\log m)$ lower bound. Our results hold for both chores with nonnegative costs and goods with nonnegative valuations.