Dynamic Spectrum Management: A Complete Complexity Characterization
Abstract: Consider a multi-user user multi multi-carrier carrier communication system where multiple users share multiple discrete subcarriers. To achieve high spectrum efficiency, the users in the system must choose their transmit power dynamically in response to fast channel fluctuations. Assuming perfect channel state information, two formulations for the spectrum management (power control) problem are considered in this paper: the first is to minimize the total transmission power subject to all users' transmission data rate constraints, and the second is to maximize the min-rate rate utility subject to individual power constraints at each user. It is known in the literature that both formulations of the prob problem lem are polynomial time solvable when the number of subcarriers is one and strongly NP-hard NP when the number of subcarriers are greater than or equal to three. However, the complexity characterization of the problem when the number of subcarriers is two has been missing for a long time. This paper answers this long long--standing open question: both formulations of the problem are strongly NP NP-hard hard when the number of subcarriers is two.