Almost Envy-Free Allocations with Connected Bundles

TL;DR

This paper investigates the existence of envy-free up to one good (EF1) allocations with connected bundles in a graph, providing existence results, algorithms, and characterizations for various cases.

Contribution

It establishes new existence results for EF1 and EF2 allocations with connected bundles, develops polynomial algorithms, and characterizes graphs guaranteeing EF1 for two agents.

Findings

EF1 exists for up to four agents on path graphs with monotonic utilities.

EF2 exists for any number of agents on path graphs.

A polynomial-time algorithm computes EF1 allocations for identical utilities.

Abstract

We study the existence of allocations of indivisible goods that are envy-free up to one good (EF1), under the additional constraint that each bundle needs to be connected in an underlying item graph. If the graph is a path and the utility functions are monotonic over bundles, we show the existence of EF1 allocations for at most four agents, and the existence of EF2 allocations for any number of agents; our proofs involve discrete analogues of the Stromquist's moving-knife protocol and the Su--Simmons argument based on Sperner's lemma. For identical utilities, we provide a polynomial-time algorithm that computes an EF1 allocation for any number of agents. For the case of two agents, we characterize the class of graphs that guarantee the existence of EF1 allocations as those whose biconnected components are arranged in a path; this property can be checked in linear time.