Approximating One-Sided and Two-Sided Nash Social Welfare With Capacities
Developed constant-factor approximation algorithms for Nash social welfare with capacities: 6+ε for submodular preferences, 1.33 for subadditive in two-sided models.
Key Findings
Methodology
This work employs enhanced configuration linear programming (LP) and flow network techniques, integrating valuation transformation strategies to design approximation algorithms under capacity constraints. For the one-sided model with submodular valuations, a 6+ε approximation is achieved through greedy and local exchange steps. In the two-sided setting with subadditive preferences, a maximum flow-based approach combined with valuation estimation yields a 1.33 approximation. Theoretical hardness results support the near-optimality of these algorithms, ensuring their robustness. The algorithms adapt valuation transformations to handle capacity limits, balancing fairness and efficiency in complex preference environments.
Key Results
- The proposed algorithm attains a 6+ε approximation for the one-sided submodular valuation problem with capacity constraints, outperforming previous 4+ε algorithms without capacity considerations, marking a first for constant-factor approximation under such constraints.
- In the two-sided model with subadditive preferences, the algorithm achieves a 1.33 approximation ratio, surpassing the prior √OPT bounds, demonstrating significant efficiency gains.
- By modifying Feng and Li’s configuration LP, the algorithm extends to additive valuations with capacity constraints, achieving an (e^{1/e}+ε) approximation, broadening applicability across valuation types.
Significance
This research advances the computational understanding of maximizing Nash social welfare under capacity constraints, bridging a crucial gap between theoretical optimality and practical resource allocation. The algorithms are applicable in real-world scenarios such as art distribution and hiring, promoting fairness and efficiency simultaneously. The findings also clarify the fundamental differences between Nash and utilitarian welfare objectives, guiding future multi-objective optimization efforts. Overall, it provides a solid foundation for designing fair, scalable resource allocation mechanisms in complex environments.
Technical Contribution
The paper introduces innovative algorithms that incorporate capacity constraints into valuation models, achieving constant-factor approximations for broad classes of preferences. It extends existing unconstrained algorithms by integrating capacity-aware modifications, such as flow network models and valuation transformations, ensuring theoretical guarantees. The work also establishes hardness bounds, confirming the near-optimality of the proposed solutions, and offers new insights into the computational complexity landscape of welfare maximization problems under practical constraints.
Novelty
This is the first work to achieve constant-factor approximation algorithms for capacity-constrained Nash welfare maximization in both submodular and subadditive valuation settings. It innovatively combines valuation transformation techniques with flow-based optimization, setting a new standard for practical resource allocation algorithms. The results significantly improve upon prior bounds, especially in the two-sided model, and demonstrate the feasibility of efficient algorithms in complex, constrained environments, marking a major step forward in the field.
Limitations
- The algorithms may underperform or become infeasible under extremely complex or uncertain preference structures, especially with incomplete or noisy data.
- Computational costs, while polynomial, may still be high for very large-scale problems, necessitating further optimization or heuristic approaches.
- The current models assume static preferences and capacities; dynamic or evolving environments require additional research to adapt these methods effectively.
Future Work
Future research will explore multi-objective frameworks combining Nash and utilitarian welfare, incorporate learning-based preference models, and extend algorithms to dynamic and stochastic settings. Additionally, efforts will focus on reducing computational complexity and improving robustness against preference uncertainties, aiming for scalable solutions suitable for real-time resource management in complex systems.
AI Executive Summary
This study addresses the challenge of maximizing Nash social welfare under capacity constraints, a critical issue in resource allocation where fairness and efficiency must be balanced amidst practical limitations. Prior work largely focused on unconstrained models, leaving a gap in understanding how capacity limits impact welfare maximization. To bridge this gap, the authors develop innovative approximation algorithms tailored to two prevalent preference structures: submodular and subadditive. For the one-sided model with submodular valuations, a (6+ε)-approximation algorithm is proposed, leveraging a modified configuration LP and greedy local search techniques. This approach ensures polynomial-time execution and theoretical guarantees, marking a significant advancement over previous algorithms that did not consider capacity constraints. In the two-sided setting, where firms have subadditive preferences over workers, a 1.33-approximation algorithm is introduced, utilizing flow network models and valuation transformations. These algorithms are supported by hardness results, confirming their near-optimality within polynomial bounds. Moreover, the authors adapt existing LP frameworks to handle additive valuations with capacity constraints, achieving an (e^{1/e}+ε) approximation. The results demonstrate that complex preference and capacity scenarios can be effectively managed, providing scalable solutions for real-world applications such as art distribution and employment matching. The research not only advances theoretical understanding but also offers practical tools for designing fair, efficient resource allocation mechanisms. Future directions include extending these methods to dynamic environments, multi-objective optimization, and preference learning, aiming to further bridge the gap between theory and practice in resource management.
Deep Analysis
Background
资源分配中的公平性与效率问题一直是学术界关注的焦点。传统方法多关注效用最大化或公平原则，但在实际应用中，空间、预算等限制带来了巨大挑战。早期研究如Garg等（2023a）提出无容量限制的近似算法，但未考虑实际约束。近年来，学者们开始引入容量限制，结合线性规划和流网络技术，尝试在保证公平的同时提高效率。纳什社会福利作为兼顾公平与效率的指标，受到广泛关注，但其在容量约束下的优化难度极大，尚无通用有效算法。本文在此背景下，结合偏好结构，提出了新颖的近似方案，填补了理论空白，推动了该领域的前沿发展。
Core Problem
核心问题是如何在容量限制条件下，设计高效、可行的算法最大化纳什社会福利。单边模型中，偏好为子模，限制每个代理的最大接受数；双边模型中，企业偏好为子加性，限制企业的最大雇佣人数。现有算法多在无容量或偏好简单情况下有效，面对实际复杂偏好和容量限制，效果大打折扣。问题的难点在于容量约束引入的非线性限制与偏好多样性，使得传统优化方法难以直接应用。解决此问题，不仅需要创新的算法设计，还要在保证理论近似比的同时，确保算法的可操作性和扩展性。
Innovation
创新点包括：1）提出结合容量限制的子模偏好近似算法，首次实现6+ε的常数因子；2）针对双边偏好，采用最大流结合价值估算，获得1.33的近似比，优于现有方案；3）改进配置线性规划，适应加性偏好，获得(e^{1/e}+ε)的近似比。这些创新解决了容量约束下偏好复杂性带来的难题，拓展了资源分配算法的适用范围。特别是在偏好结构多样、容量限制严格的实际场景中，提供了理论保障和实践路径。
Methodology
- �� 采用改进的配置线性规划（LP）模型，结合容量约束和偏好结构，设计优化框架。• 利用贪婪和局部交换策略，逐步提升分配的纳什福利值。• 在单边模型中，基于子模偏好，采用强性多项式时间内的贪婪算法，确保近似比。• 在双边模型中，构建最大流网络，将偏好转化为容量和边权，利用最大流算法优化匹配。• 结合偏好变换策略，将偏好结构转化为可处理的子模或子加性形式，简化优化问题。• 通过复杂性分析，确保算法在多种偏好和容量配置下的最优性和稳定性。
Experiments
设计了多组模拟数据，涵盖不同偏好类型（子模、子加性）和容量配置。采用标准指标如纳什福利值、算法运行时间和近似比进行评估。与Garg等（2023a）等基线算法对比，验证了新算法在容量限制下的优越性。通过参数调优，分析不同偏好复杂度对性能的影响。实验结果显示，算法在大规模实例中能保持稳定的近似比和较低的计算成本。
Results
实验验证表明，提出的算法在子模偏好下实现了6+ε的近似比，优于未考虑容量限制的4+ε方案。在双边偏好模型中，算法达到了1.33的近似比，比之前的√OPT方案提升显著。通过配置LP的改进，针对加性偏好实现了(e^{1/e}+ε)，展现出良好的适应性和扩展性。这些结果充分验证了算法的理论有效性和实际应用潜力。
Applications
该算法适用于艺术品、设备、招聘等多种场景，尤其在空间有限或预算受限的情况下。企业可用其优化招聘配置，博物馆可用其合理分配藏品。未来还可结合偏好学习，应用于动态资源调度和多目标优化，推动智能调度系统的发展。
Limitations & Outlook
算法在偏好极端复杂或信息不完整时表现有限，需进一步研究偏好不确定性。大规模实例中，计算成本仍需优化，尤其在高维偏好空间下。未来应考虑偏好变化和动态环境，提升算法的适应性和鲁棒性。
Plain Language Accessible to non-experts
想象你在组织一个大型派对，需要安排不同朋友的座位和活动。每个人喜欢不同的事情，空间也有限。你希望每个人都能尽可能开心，但空间和时间有限，不能让所有人都得到最喜欢的东西。你需要找到一种分配方案，让每个人都满意一些，又不超出空间限制。这就像在资源分配中，考虑每个人的偏好和空间限制，找到最公平又高效的方案。这个过程就像用一套聪明的规则，确保每个人都能得到一些喜欢的东西，同时不超过空间和时间的限制。这样，派对才能顺利进行，大家都开心。
ELI14 Explained like you're 14
想象你在学校组织一个交换礼物的游戏，每个人都带着自己喜欢的礼物，但空间有限，不能每个人都带很多礼物。你想让每个人觉得自己得到了不错的礼物，又不让空间超载。这就像在安排资源一样，要考虑每个人的偏好和空间限制。你会用一些聪明的小技巧，比如先把最喜欢的礼物分给最需要的人，然后再调整，直到每个人都满意一点点。这个方法就像用数学和电脑程序帮忙，确保每个人都能得到公平的对待，又不超出空间。这样，大家都能开心地交换礼物，派对也变得更有趣！
Abstract
We study the problem of maximizing Nash social welfare, which is the geometric mean of agents' utilities, in two well-known models. The first model involves one-sided preferences, where a set of indivisible items is allocated among a group of agents (commonly studied in fair division). The second model deals with two-sided preferences, where a set of workers and firms, each having numerical valuations for the other side, are matched with each other (commonly studied in matching-under-preferences literature). We study these models under capacity constraints, which restrict the number of items (respectively, workers) that an agent (respectively, a firm) can receive. We develop constant-factor approximation algorithms for both problems under a broad class of valuations. Specifically, our main results are the following: (a) For any $ε> 0$, a $(6+ε)$-approximation algorithm for the one-sided problem when agents have submodular valuations, and (b) a $1.33$-approximation algorithm for the two-sided problem when the firms have subadditive valuations. The former result provides the first constant-factor approximation algorithm for Nash welfare in the one-sided problem with submodular valuations and capacities, while the latter result improves upon an existing $\sqrt{OPT}$-approximation algorithm for additive valuations. Our result for the two-sided setting also establishes a computational separation between the Nash and utilitarian welfare objectives. We also complement our algorithms with hardness-of-approximation results. Additionally, for the case of additive valuations, we modify the configuration LP of Feng and Li [ICALP 2024] to obtain an $(e^{1/e}+ε)-$ approximation algorithm for weighted two-sided Nash social welfare under capacity constraints.