
Extension of Additive Valuations to General Valuations on the Existence of EFX
Envyfreeness is one of the most widely studied notions in fair division...
read it

Fair division of graphs and of tangled cakes
A tangle is a connected topological space constructed by gluing several ...
read it

One Dollar Each Eliminates Envy
We study the fair division of a collection of m indivisible goods amongs...
read it

Simple Necessary Conditions for the Existence of a Hamiltonian Path with Applications to Cactus Graphs
We describe some necessary conditions for the existence of a Hamiltonian...
read it

Fair Division via Social Comparison
In the classical cake cutting problem, a resource must be divided among ...
read it

A Discrete and Bounded EnvyFree Cake Cutting Protocol for Any Number of Agents
We consider the wellstudied cake cutting problem in which the goal is t...
read it

Online Cake Cutting (published version)
We propose an online form of the cake cutting problem. This models situa...
read it
Almost EnvyFree Allocations with Connected Bundles
We study the existence of allocations of indivisible goods that are envyfree up to one good (EF1), under the additional constraint that each bundle needs to be connected in an underlying item graph G. When the items are arranged in a path, we show that EF1 allocations are guaranteed to exist for arbitrary monotonic utility functions over bundles, provided that either there are at most four agents, or there are any number of agents but they all have identical utility functions. Our existence proofs are based on classical arguments from the divisible cakecutting setting, and involve discrete analogues of cutandchoose, of Stromquist's movingknife protocol, and of the SuSimmons argument based on Sperner's lemma. Sperner's lemma can also be used to show that on a path, an EF2 allocation exists for any number of agents. Except for the results using Sperner's lemma, all of our procedures can be implemented by efficient algorithms. Our positive results for paths imply the existence of connected EF1 or EF2 allocations whenever G is traceable, i.e., contains a Hamiltonian path. For the case of two agents, we completely characterize the class of graphs G that guarantee the existence of EF1 allocations as the class of graphs whose biconnected components are arranged in a path. This class is strictly larger than the class of traceable graphs; one can be check in linear time whether a graph belongs to this class, and if so return an EF1 allocation.
READ FULL TEXT
Comments
There are no comments yet.