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

2026-07-10

AI Summary

This work addresses the problem of envy-free allocation of indivisible items—comprising goods, chores, or a mixture thereof—among multiple agents with additive valuations. It presents the first subexponential-time algorithm for this setting by modeling feasible allocations as a convex polytope in ℝ³ and recursively partitioning the agent set using Miller’s planar separator theorem. The resulting divide-and-conquer framework integrates geometric and graph-theoretic techniques to determine, in $(n \cdot m)^{O(\sqrt{n})}$ time, whether an envy-free allocation exists and to construct such an allocation when it does. The algorithm also accommodates instances where parts of the allocation are pre-fixed, thereby enabling the first efficient treatment of high-multiplicity mixed instances.

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.

Problem

• envy-free allocation
• indivisible items
• mixed instances
• fair division
• subexponential algorithm

Innovation

• subexponential algorithm
• envy-free allocation
• mixed goods and chores
• geometric representation
• cycle-separator theorem