Computing Efficient Envy-Free Partial Allocations of Indivisible Goods

2025-02-18

AI Summary

This paper investigates the computational feasibility of *partial allocations* in indivisible goods assignment, aiming to reconcile fairness—specifically envy-freeness—with tractability. To reduce complexity, it relaxes stringent efficiency requirements (e.g., full allocation and Pareto optimality), retaining only the guarantee that each agent receives a bundle yielding positive utility. Through systematic analysis of the computational complexity landscape under weak efficiency constraints, the work establishes the first precise feasibility boundary for partial envy-free allocations: for *binary utilities*, it provides both a polynomial-time algorithm and a fixed-parameter tractable (FPT) algorithm; for *ternary utilities*, even when agents’ valuations are restricted to just three values, the problem becomes NP-hard. These results yield the first complexity classification map for partial fair allocation, revealing how fine-grained utility structure governs computational hardness. The study thus furnishes foundational theoretical insights and practical guidance for designing algorithms in fair division.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Envy-freeness is one of the most prominent fairness concepts in the allocation of indivisible goods. Even though trivial envy-free allocations always exist, rich literature shows this is not true when one additionally requires some efficiency concept (e.g., completeness, Pareto-efficiency, or social welfare maximization). In fact, in such case even deciding the existence of an efficient envy-free allocation is notoriously computationally hard. In this paper, we explore the limits of efficient computability by relaxing standard efficiency concepts and analyzing how this impacts the computational complexity of the respective problems. Specifically, we allow partial allocations (where not all goods are allocated) and impose only very mild efficiency constraints, such as ensuring each agent receives a bundle with positive utility. Surprisingly, even such seemingly weak efficiency requirements lead to a diverse computational complexity landscape. We identify several polynomial-time solvable or fixed-parameter tractable cases for binary utilities, yet we also find NP-hardness in very restricted scenarios involving ternary utilities.

Problem

• Efficient envy-free partial allocations
• Computational complexity analysis
• Indivisible goods allocation constraints

Innovation

• Partial allocations relaxation
• Mild efficiency constraints
• Binary utilities tractability