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

11/04/2022
by   Ganesh Ghalme, et al.
0

We study the classic problem of fairly dividing a heterogeneous and divisible resource – modeled as a line segment [0,1] and typically called as a 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 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 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.

READ FULL TEXT
research
05/25/2022

Envy-Free Cake Cutting with Graph Constraints

We study the classic problem of fairly dividing a heterogeneous and divi...
research
07/11/2019

Cake Cutting on Graphs: A Discrete and Bounded Proportional Protocol

The classical cake cutting problem studies how to find fair allocations ...
research
11/20/2016

Fair Division via Social Comparison

In the classical cake cutting problem, a resource must be divided among ...
research
06/06/2023

An Exponential Envy-Free Cake Cutting Protocol for n Agents

We consider a classical envy-free cake cutting problem. The first limite...
research
06/21/2021

Thou Shalt Covet The Average Of Thy Neighbors' Cakes

We prove an Ω(n^2) lower bound on the query complexity of local proporti...
research
04/13/2016

A Discrete and Bounded Envy-Free Cake Cutting Protocol for Any Number of Agents

We consider the well-studied cake cutting problem in which the goal is t...
research
07/01/2018

An Improved Envy-Free Cake Cutting Protocol for Four Agents

We consider the classic cake-cutting problem of producing envy-free allo...

Please sign up or login with your details

Forgot password? Click here to reset