A Discrete and Bounded Locally Envy-Free Cake Cutting Protocol on Trees

TL;DR

This paper introduces a simple, discrete, and bounded protocol for locally envy-free cake division on trees, significantly reducing query complexity compared to previous methods, especially for shallow trees.

Contribution

It presents the first polynomial-query protocol for locally envy-free cake cutting on certain non-trivial graph structures, specifically trees with limited depth.

Findings

Bounded protocol with exponential queries for general trees.

Polynomial query protocol for trees with depth at most two.

Improved query complexity over previous hyper-exponential bounds.

Abstract

We study the classic problem of \emph{fairly} dividing a heterogeneous and divisible resource -- modeled as a line segment $[0,1]$ and typically called as a \emph{cake} -- among $n$ agents. This work considers an interesting variant of the problem where agents are embedded on a graph. The graphical constraint entails that each agent evaluates her allocated share only against her neighbors' share. Given a graph, the goal is to efficiently find a \emph{locally envy-free} allocation where every agent values her share of the cake to be at least as much as that of any of her neighbors' share. The most significant contribution of this work is a bounded protocol that finds a locally envy-free allocation among $n$ agents on a tree graph using $n^{O(n)}$ queries under the standard Robertson-Webb (RW) query model. The query complexity of our proposed protocol, though exponential, significantly improves the currently best known hyper-exponential query complexity bound of Aziz and Mackenzie [AM16] for complete graphs. In particular, we also show that if the underlying tree graph has a depth of at most two, one can find a locally envy-free allocation with $O(n^4 \log n)$ RW queries. This is the first and the only known locally envy-free cake cutting protocol with polynomial query complexity for a non-trivial graph structure. Interestingly, our discrete protocols are simple and easy to understand, as opposed to highly involved protocol of [AM16]. This simplicity can be attributed to their recursive nature and the use of a single agent as a designated \emph{cutter}. We believe that these results will help us improve our algorithmic understanding of the arguably challenging problem of envy-free cake-cutting by uncovering the bottlenecks in its query complexity and its relation to the underlying graph structures.