Dynamic Spectrum Management: A Complete Complexity Characterization

TL;DR

This paper thoroughly analyzes the computational complexity of dynamic spectrum management in multi-user, multi-carrier systems, revealing that the problem is strongly NP-hard even with only two subcarriers, resolving a long-standing open question.

Contribution

It provides a complete complexity characterization of spectrum management problems, showing they are strongly NP-hard for two or more subcarriers, filling a key gap in the literature.

Findings

Both formulations are strongly NP-hard with two subcarriers.

The problem is polynomial-time solvable when there is only one subcarrier.

Complexity remains high even with minimal subcarrier count.

Abstract

Consider a multi-user multi-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 utility subject to individual power constraints at each user. It is known in the literature that both formulations of the problem are polynomial time solvable when the number of subcarriers is one and strongly NP-hard 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-standing open question: both formulations of the problem are strongly NP-hard when the number of subcarriers is two.