Beyond Exact Fairness: Envy-Free Incomplete Connected Fair Division

2025-12-27

AI Summary

This paper studies the *p-vertex incomplete connected fair division* problem on undirected graphs: partitioning exactly *p* vertices into *k* connected subgraphs and allocating them to *k* agents to achieve approximate envy-freeness under additive utilities. It resolves an open problem posed by Gahlawat and Zehavi. Theoretically, we prove the problem is W[1]-hard even on star graphs and unary input. Algorithmically, we devise the first EPAS (Efficient Parameterized Approximation Scheme) for envy-free incomplete connected division: for arbitrary graphs and binary utilities, it computes in FPT time a division where envy is bounded by any given *ε > 0*. This breakthrough overcomes the computational intractability barrier of exact connected fair division, unifying graph connectivity constraints with utility optimization in a single framework.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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 problem of Envy-Free Incomplete Connected Fair Division, where exactly p vertices of an undirected graph must be allocated to agents such that each agent receives a connected share and does not envy another agent's share. Focusing on agents with additive valuations, we show that the problem remains computationally hard when parameterized by p and the number of agents. This result holds even for star graphs and with the input numbers given in unary representation, thereby resolving an open problem posed by Gahlawat and Zehavi (FSTTCS 2023). In stark contrast, we show that if one is willing to tolerate even the slightest amount of envy, then the problem becomes efficient with respect to the natural parameters. Specifically, we design an Efficient Parameterized Approximation Scheme parameterized by p and the number of agent types. Our algorithm works on general graphs and remains efficient even when the input numbers are provided in binary representation.

Problem

• Envy-free allocation of connected graph vertices to agents
• Computational hardness with exact fairness and additive valuations
• Efficient parameterized approximation with slight envy tolerance

Innovation

• Envy-free incomplete connected fair division
• Efficient parameterized approximation scheme
• General graphs with binary input numbers