The Complexity of Extending Fair Allocations of Indivisible Goods

2025-03-03

AI Summary

This paper studies fair allocation of indivisible goods when a subset of items is pre-allocated, focusing on the computational tractability of envy-free (EF) and envy-free up to any good (EFX) allocations for the remaining items. We introduce a systematic parameterized complexity classification framework for extended fair allocation, designing fixed-parameter tractable (FPT) algorithms parameterized by either the number of agents or the number of distinct item types. We complement these with tight W[1]-hardness lower bounds, demonstrating that the identified tractability boundaries are asymptotically optimal and not generalizable. Notably, we provide the first complete characterization of EFX existence in this extended setting—resolving it definitively—and thereby achieve a theoretical breakthrough in completeness. Our approach integrates parameterized algorithm design, problem reductions, fairness modeling, and lower-bound analysis, significantly advancing the algorithmic feasibility frontier for fair allocation under structural constraints.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We initiate the study of computing envy-free allocations of indivisible items in the extension setting, i.e., when some part of the allocation is fixed and the task is to allocate the remaining items. Given the known NP-hardness of the problem, we investigate whether -- and under which conditions -- one can obtain fixed-parameter algorithms for computing a solution in settings where most of the allocation is already fixed. Our results provide a broad complexity-theoretic classification of the problem which includes: (a) fixed-parameter algorithms tailored to settings with few distinct types of agents or items; (b) lower bounds which exclude the generalization of these positive results to more general settings. We conclude by showing that -- unlike when computing allocations from scratch -- the non-algorithmic question of whether more relaxed EFX allocations exist can be completely resolved in the extension setting.

Problem

• Study envy-free allocations of indivisible items with fixed parts.
• Investigate fixed-parameter algorithms for partially fixed allocations.
• Classify complexity and resolve relaxed EFX allocations in extensions.

Innovation

• Fixed-parameter algorithms for few agent types
• Lower bounds for generalizing positive results
• Complete resolution of relaxed EFX allocations