Complexity results for the Pilot Assignment problem in Cell-Free Massive MIMO

TL;DR

This paper proves that the Pilot Assignment problem in Cell-Free Massive MIMO systems is computationally hard, showing it is strongly NP-hard and cannot be approximated within certain bounds, highlighting fundamental limitations in resource allocation.

Contribution

It establishes the NP-hardness of the Pilot Assignment problem in CF-mMIMO and provides approximation bounds, advancing understanding of its computational complexity.

Findings

PA is strongly NP-hard.

No polynomial-time constant-factor approximation exists for PA.

Approximation bounds of 1.058 and ε|K|^2 are established for special cases.

Abstract

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.