On-line scheduling of unit time jobs with rejection: Minimizing the total completion time

We consider on-line scheduling of unit time jobs on a single machine with job-dependent penalties. The jobs arrive on-line (one by one) and can be either accepted and scheduled, or be rejected at the cost of a penalty. The objective is to minimize the total completion time of the accepted jobs plus the sum of the penalties of the rejected jobs. We give an on-line algorithm for this problem with competitive ratio 1/2(2 + √(3)) ≈ 1.86602. Moreover, we prove that there does not exist an on-line algorithm with competitive ratio better than 1.63784.