Minimizing the Cost of EFx Allocations

2026-01-16

AI Summary

This study addresses the problem of resource allocation that simultaneously satisfies EFx (envy-freeness up to any good) fairness and minimizes total cost, formally introducing the minCost-EFx Allocation problem for the first time. Through parameterized complexity analysis, kernelization techniques, and combinatorial optimization, the authors prove that the problem is NP-hard and inapproximable within any factor ρ unless P = NP, even in the case of just two agents, thereby establishing its intrinsic computational hardness. Nevertheless, for restricted settings—such as bounded valuations, a constant number of agents, or specific cost structures—the work identifies polynomial kernels and delineates several tractable special cases, thus bridging a critical theoretical gap in the joint optimization of fairness and efficiency.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Ensuring fairness while limiting costs, such as transportation or storage, is an important challenge in resource allocation, yet most work has focused on cost minimization without fairness or fairness without explicit cost considerations. We introduce and formally define the minCost-EFx Allocation problem, where the objective is to compute an allocation that is envy-free up to any item (EFx) and has minimum cost. We investigate the algorithmic complexity of this problem, proving that it is NP-hard already with two agents. On the positive side, we show that the problem admits a polynomial kernel with respect to the number of items, implying that a core source of intractability lies in the number of items. Building on this, we identify parameter-restricted settings that are tractable, including cases with bounded valuations and a constant number of agents, or a limited number of item types under restricted cost models. Finally, we turn to cost approximation, proving that for any $\rho>1$ the problem is not $\rho$-approximable in polynomial time (unless $P=NP$), while also identifying restricted cost models where costs are agent-specific and independent of the actual items received, which admit better approximation guarantees.

Problem

• EFx
• cost minimization
• resource allocation
• fairness
• NP-hard

Innovation

• EFx allocation
• cost minimization
• parameterized complexity
• polynomial kernel
• approximation hardness