We address a generalization of the classical discrete time-cost tradeoff problem where the costs are irregular and depend on the starting and the completion times of the activities. We present a complete picture of the computational complexity and the approximability of this problem for several natural classes of precedence constraints. We prove that the problem is np-hard and hard to approximate, even in case the precedence constraints form an interval order. For precedence constraints with bounded height, there is a complexity jump from height one to height two: for height one, the problem is polynomially solvable, whereas for height two, it is np-hard and apx-hard. Finally, the problem is shown to be polynomially solvable if the precedence constraints have bounded width or are series parallel.
Grigoriev, A., & Woeginger, G. J. (2004). Project scheduling with irregular costs: complexity, approximability, and algorithms. Acta Informatica, 41(2-3), 83-97. https://doi.org/10.1007/s00236-004-0150-2