On-Line Single Machine Scheduling of Unit Time Jobs with Rejection: Minimizing the Maximum Quadratic Completion Time

In this paper, we consider the on-line single machine scheduling of unit time jobs with rejection. All jobs arrive on-line over a list (one by one). For each arriving job, the on-line algorithm must decide immediately to accept or reject it. The objective is to minimize the maximum quadratic completion time of accepted jobs plus the total rejection cost of rejected jobs. For this problem, we show that 1.7299 is a lower bound on the competitive ratio and present a simple greedy algorithm with the competitive ratio 2. Furthermore, we also provide a modified greedy algorithm with a better competitive ratio 1 + root 3/2 approximate to 1.86602.