  Consider the cellular downlink Orthogonal Frequency Division Multiple Access
(OFDMA) system where a single transmitter transmits signals to multiple
receivers on multiple discrete subcarriers. To adapt fast channel fluctuations,
the transmitter should be able to dynamically allocate subcarrier and power
resources. Assuming perfect channel knowledge, we formulate the joint
subcarrier and power allocation problem as two optimization problems: the first
is the one of minimizing the total transmission power subject to quality of
service constraints, and the second is the one of maximizing a system utility
function subject to power budget constraints. In this letter, we show that both
the aforementioned formulations of the joint subcarrier and power allocation
problem are generally NP-hard. We also identify several subclasses of the
problem which are polynomial time solvable.
