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arXiv · 2607.09327

Subexponential Algorithm for High Multiplicity Fair Division of Mixed Instances via Stereometry

Yuriy Dementiev·Fedor Pribytkov·Danil Sagunov
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Abstract

We study the problem of computing an envy-free (EF) allocation of $m$ indivisible items among $n$ agents when items come in three distinct types. Each agent holds additive valuations over item types that may be positive (goods), negative (chores), or mixed. We present the first subexponential-time algorithm with running time time $(n \cdot m)^{O(\sqrt{n})}$ that finds an EF allocation whenever one exists, or correctly reports that none exists. Our approach exploits a geometric representation of EF allocations as convex polyhedra in $\mathbb{R}^3$ and applies Miller's planar cycle-separator theorem to recursively decompose the agent set into balanced subgroups. We further extend the algorithm to handle agents whose allocations are fixed in advance, preserving envy-freeness across all agents.

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Yuriy Dementiev, Fedor Pribytkov, Danil Sagunov. 2026-07-10. Subexponential Algorithm for High Multiplicity Fair Division of Mixed Instances via Stereometry. https://arxiv.org/abs/2607.09327

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