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：无线通信正在使数十亿人相互连接并接入互联网，推动经济各领域的转型，并构建强大新技术的基础，这些技术有望以前所未有的速度和规模改善人类生活。设备数量的急剧增长以及对更高数据速率和更广网络覆盖的相关需求，推动了对更稳健无线技术的需求。解决这一问题的一项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，其中之一是高效分配有限资源。本文聚焦于无线网络中的一个主要资源分配问题，即导频分配问题。我们证明导频分配问题是强NP-困难的，并且不存在多项式时间的常数因子近似算法。此外，我们证明当系统包含至少三个导频时，导频分配问题无法在多项式时间内以$\mathcal{O}(K^2)$（其中$K$为用户数）的近似比进行近似。最后，在系统恰好包含两个（或三个）导频的特殊情形下，我们分别给出了$1.058$（或$\epsilon|K|^2$，其中$\epsilon>0$）的近似下界。