Single machine scheduling with release dates

We consider the scheduling problem of minimizing the average weighted completion time of n jobs with release dates on a single machine. We first study two linear programming relaxations of the problem, one based on a time-indexed formulation, the other on a completion-time formulation. We show their equivalence by proving that a O(n log n) greedy algorithm leads to optimal solutions to both relaxations. The proof relies on the notion of mean busy times of jobs, a concept which enhances our understanding of these LP relaxations. Based on the greedy solution, we describe two simple randomized approximation algorithms, which are guaranteed to deliver feasible schedules with expected objective function value within factors of 1.7451 and 1.6853, respectively, of the optimum. They are based on the concept of common and independent $\alpha$-points, respectively. The analysis implies in particular that the worst-case relative error of the LP relaxations is at most 1.6853, and we provide instances showing that it is at least $e/(e-1) \approx 1.5819$. Both algorithms may be derandomized; their deterministic versions run in O(n2) time. The randomized algorithms also apply to the on-line setting, in which jobs arrive dynamically over time and one must decide which job to process without knowledge of jobs that will be released afterwards.