Heat exchanger network synthesis (HENS) has been the subject of a significant amount of research over the last 40 years. While significant progress has been made towards solving the problem, its computational complexity is not known, i.e., it is not known whether a polynomial algorithm might exist for the problem or not. This issue is addressed in this paper through a computational complexity analysis. We prove that HENS is, NP-hard, thus refuting the possibility for the existence of a computationally efficient (polynomial) exact solution algorithm for this problem. While this complexity characterization may not be surprising, our analysis shows that HENS is NP-hard in the strong sense. Therefore, HENS belongs to a particularly difficult class of hard optimization problems. Further, via restriction to the 3-partition problem, our complexity proofs reveal that even the following simple HENS subproblems are. NP-hard in the strong sense: (a) the minimum number of matches target problem, (b) the matches problem with only one temperature interval, uniform cost coefficients, and uniform heat requirements of all cold streams. These results facilitate the computational complexity analysis of more complex HENS problems and provide new insights to structural properties of the problem. They also provide motivation for the development of specialized optimization algorithms and approximation schemes.