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Consider a multi-user orthogonal frequency division multiple access (OFDMA) system where multiple users share multiple discrete subcarriers, but at most one user is allowed to transmit power on each subcarrier. To adapt fast traffic and channel fluctuations and improve the spectrum efficiency, the system should have the ability to dynamically allocate subcarriers and power resources to users. Assuming perfect channel knowledge, two formulations for the joint subcarrier and power allocation problem are considered in this paper: the first is to minimize the total transmission power subject to the quality of service constraints and the OFDMA constraint, and the second is to maximize some system utility function subject to the total transmission power constraint per user and the OFDMA constraint. In spite of the existence of various heuristics approaches, little is known about the computational complexity status of the above problem. This paper aims at filling this theoretical gap, i.e., characterizing the complexity of the joint subcarrier and power allocation problem for the multi-user OFDMA system. It is shown in this paper that both formulations of the joint subcarrier and power allocation problem are strongly NP-hard. Several subclasses of the problem which can be solved efficiently in polynomial time are also identified. These complexity results suggest that there are not polynomial time algorithms that are able to solve the general joint subcarrier and power allocation problem to global optimality (unless P=NP), and determining an approximately optimal subcarrier and power allocation strategy is more realistic in practice.