Scheduling with Rejection to Minimize the Makespan

In this paper, we consider the scheduling with rejection. The objective functions are to minimize the maximum completion time of the processed ones when the total compression cost is given. Firstly, we prove that the problem 1 vertical bar rej vertical bar C(max)/TCP is NP-hard, which implying that P(m)vertical bar rej vertical bar C(max)/TCP, 1 vertical bar rej, r(j)vertical bar C(max)/TCP, 1 vertical bar rej, on - line vertical bar C(max)/TCP are all NP-hard. Secondly, for problem P(m)vertical bar rej vertical bar C(max)/TCP we design a pseudopolynomial time dynamic programming algorithm that solves it exactly and an FPTAS (full polynomial time approximation scheme) when,in is a constant. We also design a pseudopolynomial time dynamic programming algorithm and an FPTAS for the case of non-identical job arrival problem 1 vertical bar rej, r(j)vertical bar C(max)/TCP. In the end, we consider the on-line problem 1 vertical bar rej, on - line vertical bar C(max)/TCP and prove that there doesn't exist any on-line algorithm with a constant competitive ratio for it, even if the jobs only have two different release times.