Graphical Cake Cutting via Maximin Share

TL;DR

This paper explores fair division of a graph-represented cake, proving maximin share fairness for forests and discussing relaxations for general graphs, with implications for road network divisions.

Contribution

It introduces the first results on maximin share fairness in graph-based cake-cutting, especially for forests and general graphs, including relaxations.

Findings

Maximin share fairness always exists for forest graphs.

No proportionality approximation is possible with separation constraints.

Ordinal relaxations are achievable for general graphs.

Abstract

We study the recently introduced cake-cutting setting in which the cake is represented by an undirected graph. This generalizes the canonical interval cake and allows for modeling the division of road networks. We show that when the graph is a forest, an allocation satisfying the well-known criterion of maximin share fairness always exists. Our result holds even when separation constraints are imposed, in which case no multiplicative approximation of proportionality can be guaranteed. Furthermore, while maximin share fairness is not always achievable for general graphs, we prove that ordinal relaxations can be attained.