From Cake-Cutting and Necklace-Splitting to Fair Division of Indivisible Items

2026-08-04

AI Summary

This study addresses the challenge of extending continuous fair division theory—such as cake cutting and necklace splitting—to the fair allocation of indivisible items arranged on a path, under connectivity and consensus constraints. By constructing an existence-preserving transfer framework that integrates connected cake-cutting theorems with Jojić et al.'s equipartition results for necklaces, and employing combinatorial topological methods to handle settings with a prime power number of agents, the work achieves the first EFk-type fair allocations under non-additive valuations while respecting both connectivity and approximate consensus. The main contributions include overcoming prior limitations to binary partitions, establishing EF2 allocations for prime power numbers of agents with bundle sizes differing by at most two, and providing a general construction for arbitrary valuations yielding bundles composed of at most n intervals and consensus error bounded by n items.

Abstract

We give an existential transfer framework for converting continuous fair division theorems into guarantees for indivisible items arranged on a path. This allows continuous envy-freeness and consensus results to translate directly into EF$k$-type guarantees for indivisible allocations. Combining this method with connected cake-cutting theorems, we obtain connected allocations satisfying envy-freeness up to one good and one chore for identical valuations and for arbitrary valuations when the number of agents is a prime power. Combining this method with the equicardinal necklace-splitting theorem of Jojić et al., we show that, for any prime-power number $r$ of bundles and $n$ arbitrary valuation functions, there exists an allocation in which every bundle is the union of at most $n$ intervals, and the bundles satisfy consensus up to $n$ goods and $n$ chores. This result is the first EF$k$-type guarantee for consensus fair division with non-additive valuations beyond the halving case. Envy-freeness constraints can be imposed simultaneously at the cost of one additional interval and one additional item in each guarantee. As a consequence, when the number of agents is a prime power, every instance with monotone valuations admits an EF$2$ allocation whose bundle sizes differ by at most two.

Problem

• fair division
• indivisible items
• envy-freeness
• necklace-splitting
• cake-cutting

Innovation

• fair division
• indivisible items
• envy-freeness up to k items
• necklace splitting
• existential transfer framework