Single-machine scheduling problems with start-time dependent processing time

In this paper, we study the single-machine makespan scheduling problem with start-time dependent processing time. The objectives are to minimize the makespan and to minimize the sum of the kkth powers of completion times. We prove that several cases are polynomially solvable under some restrictions of the parameters. In addition, these cases still remain polynomially solvable when the restriction of a certain parameter is relaxed from a positive integer to a real number.