Dividing Conflicting Items Fairly

2025-06-17

AI Summary

This paper studies fair allocation of indivisible goods under graph constraints: goods correspond to vertices, and conflict edges prohibit assigning adjacent vertices to the same agent—thus each agent must receive an independent set. We focus on the existence and computability of allocations that maximize the number of agents satisfying EF1 (envy-freeness up to one good). We establish that a maximum EF1 allocation always exists for any graph and two monotone utility agents, but existence is not guaranteed for three or more agents—and the decision problem becomes NP-hard. We design a polynomial-time algorithm for additive utilities and a pseudopolynomial-time algorithm for general monotone utilities, integrating independent set theory with fair division analysis. Our results precisely characterize the threshold for EF1 existence under conflicts (n = 2 versus n ≥ 3), provide efficient constructive algorithms, and uniformly handle both goods and chores allocation.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Allocating indivisible goods under graph-based conflicts
• Ensuring maximal EF1 fairness for two agents
• Extending results to any graph with monotone valuations

Innovation

• Maximal EF1 allocation for any graph
• Polynomial-time algorithm for additive valuations
• Pseudo-polynomial algorithm for monotone valuations