We establish necessary and sufficient conditions for a network configuration to provide utilities that are both fair and efficient in a well-defined sense. To cover as many applications as possible with a unified framework, we consider utilities defined in an axiomatic way, and the constraints imposed on the feasible network configurations are expressed with a single inequality involving a monotone norm. In this setting, we prove that a necessary and sufficient condition to obtain network configurations that are efficient in the weak Pareto sense is to select configurations attaining equality in the monotone norm constraint. Furthermore, for a given configuration satisfying this equality, we characterize a criterion for which the configuration can be considered fair for the active links. We illustrate potential implications of the theoretical findings by presenting, for the first time, a simple parametrization based on power vectors of achievable rate regions in modern cell-less systems subject to practical impairments.

翻译：我们建立了网络配置在明确定义下同时实现公平与高效效用的必要充分条件。为通过统一框架覆盖尽可能多的应用场景，我们采用公理化方式定义效用函数，并将可行网络配置约束表示为含单调范数的单一不等式。在此设定下，我们证明实现弱帕累托意义上高效网络配置的充要条件是选择满足单调范数等式约束的配置。进一步地，针对满足该等式的给定配置，我们刻画了其可被视为对活动链路公平的判定准则。通过首次提出基于功率向量的简单参数化方法（适用于存在实际损伤的现代无蜂窝系统可达速率区域），我们展示了理论发现潜在的应用价值。