Wireless communication is enabling billions of people to connect to each other and the internet, transforming every sector of the economy, and building the foundations for powerful new technologies that hold great promise to improve lives at an unprecedented rate and scale. The rapid increase in the number of devices and the associated demands for higher data rates and broader network coverage fuels the need for more robust wireless technologies. The key technology identified to address this problem is referred to as Cell-Free Massive MIMO (CF-mMIMO). CF-mMIMO is accompanied by many challenges, one of which is efficiently allocating limited resources. In this paper, we focus on a major resource allocation problem in wireless networks, namely the Pilot Assignment problem (PA). We show that PA is strongly NP-hard and that it does not admit a polynomial-time constant-factor approximation algorithm. Further, we show that PA cannot be approximated in polynomial time within $\mathcal{O}(K^2)$ (where $K$ is the number of users) when the system consists of at least three pilots. Finally, we present an approximation lower bound of $1.058$ (resp. $\epsilon|K|^2$, for $\epsilon >0$) in special cases where the system consists of exactly two (resp. three) pilots.

翻译：无线通信正使数十亿人相互连接并接入互联网，推动经济各领域转型，并以前所未有的速度和规模为改善人类生活构建具有巨大潜力的新兴技术基础。设备数量的快速增长，以及随之而来的对更高数据速率和更广网络覆盖的需求，推动了对更强大无线技术的需求。应对该问题的关键技术被称为无蜂窝大规模MIMO（CF-mMIMO）。CF-mMIMO伴随着诸多挑战，其中之一便是高效分配有限资源。本文聚焦无线网络中的主要资源分配问题——导频分配问题（PA）。我们证明PA是强NP难问题，且不存在多项式时间常数因子近似算法。进一步，当系统包含至少三个导频时，我们证明PA不能在多项式时间内以$\mathcal{O}(K^2)$（其中$K$为用户数）的近似比进行求解。最后，在系统恰好包含两个（或三个）导频的特殊情况下，我们分别给出了$1.058$（或$\epsilon|K|^2$，其中$\epsilon >0$）的近似下界。