We consider the single-machine scheduling with coupled task and rejection, in which each coupled task is either accepted and processed on a single machine or rejected with a certain rejection penalty. Each accepted coupled task Jj is made of two tasks with processing times aj and bj, respectively, and a fixed exactly time interval Lj between the two tasks is required. The objective is to minimize the makespan of the accepted coupled task and the total penalty of the rejected coupled task. We present 3-approximation algorithm for the cases of aj=Lj=p and bj=Lj=p, respectively, and prove that the problems with common bj or common aj is polynomially solvable. Furthermore, when aj=bj=p, we provide a 2-approximation algorithm for Lj>p and show that the problem with for Lj≤p can be solved in polynomial time.
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