The maximum Nash social welfare (NSW) -- which maximizes the geometric mean of agents' utilities -- is a fundamental solution concept with remarkable fairness and efficiency guarantees. The computational aspects of NSW have been extensively studied for one-sided preferences where a set of agents have preferences over a set of resources. Our work deviates from this trend and studies NSW maximization for two-sided preferences, wherein a set of workers and firms, each having a cardinal valuation function, are matched with each other. We provide a systematic study of the computational complexity of maximizing NSW for many-to-one matchings under two-sided preferences. Our main negative result is that maximizing NSW is NP-hard even in a highly restricted setting where each firm has capacity 2, all valuations are in the range {0,1,2}, and each agent positively values at most three other agents. In search of positive results, we develop approximation algorithms as well as parameterized algorithms in terms of natural parameters such as the number of workers, the number of firms, and the firms' capacities. We also provide algorithms for restricted domains such as symmetric binary valuations and bounded degree instances.

翻译：最大化纳什社会福利（NSW）——即最大化主体效用的几何平均数——是一种具有显著公平性和效率保障的基本解概念。NSW 的计算方面已在一方偏好（即一组主体对一组资源具有偏好）情形下得到广泛研究。我们的工作偏离这一趋势，研究双边偏好下的 NSW 最大化问题，其中一组工人和企业各自具有基数估值函数，并相互匹配。我们系统研究了双边偏好下多对一匹配中最大化 NSW 的计算复杂性。我们的主要负面结果是：即使在高度受限的设置中（每个企业容量为2，所有估值在{0,1,2}范围内，且每个主体最多对三个其他主体赋予正值），最大化 NSW 仍然是 NP-hard 的。在寻找正面结果的过程中，我们开发了近似算法以及基于自然参数（如工人数量、企业数量及企业容量）的参数化算法。我们还针对对称二元估值和有界度实例等受限领域提供了算法。