Dividing Conflicting Items Fairly

TL;DR

This paper proves the existence of maximal EF1 allocations for two agents with monotone valuations under any graph conflict structure, provides algorithms for their computation, and shows the problem's computational hardness for three or more agents.

Contribution

It extends the existence result of EF1 allocations from interval graphs to all graphs for two agents and introduces algorithms for their computation, along with hardness results for more agents.

Findings

Maximal EF1 allocations exist for any graph with two agents and monotone valuations.

Polynomial-time algorithm for additive valuations to compute EF1 allocations.

NP-hardness of deciding EF1 allocations for three or more agents.

Abstract

We study the allocation of indivisible goods under conflicting constraints, represented by a graph. In this framework, vertices correspond to goods and edges correspond to conflicts between a pair of goods. Each agent is allocated an independent set in the graph. In a recent work of Kumar et al. (2024), it was shown that a maximal EF1 allocation exists for interval graphs and two agents with monotone valuations. We significantly extend this result by establishing that a maximal EF1 allocation exists for \emph{any graph} when the two agents have monotone valuations. To compute such an allocation, we present a polynomial-time algorithm for additive valuations, as well as a pseudo-polynomial time algorithm for monotone valuations. Moreover, we complement our findings by providing a counterexample demonstrating a maximal EF1 allocation may not exist for three agents with monotone valuations; further, we establish NP-hardness of determining the existence of such allocations for every fixed number $n \geq 3$ of agents. All of our results for goods also apply to the allocation of chores.