Optimal algorithms for single-machine scheduling with rejection to minimize the makespan

In this paper we consider the single-machine scheduling problem with rejection. Each job has an arrival time, a processing time and a rejection penalty. The objective is to minimize the sum of the makespan of the accepted jobs and the total rejection penalty of the rejected jobs. First, we present a polynomial-time algorithm for the off-line problem when split is allowed. Second, for the on-line problem with arbitrary arrival times, we provide a determined on-line algorithm with the best-possible competitive ratio 2. Finally, for the on-line problem with at most two distinct arrival times, we provide a determined on-line algorithm with the best-possible competitive ratio (5+1)/2≈1.618.