Improved approximation algorithms for scheduling with release dates

We consider the scheduling problem of minimizing weighted sum of completion times under release dates. We present a simple randomized 2-approximation algorithm for the problem running in O(n log n) time. The algorithm applies to both the off-line and on-line settings with no difference in performance guarantees. In the off-line setting, the algorithm can be derandomized in two different ways, leading either to a deterministic 5-approximation algorithm running in O(n(2)) time, or to a 2+1/k-approximation algorithm running in O(n log n+kn) time. In the on-line setting, a specific random choice gives a deterministic 1 + root 2-competitive algorithm. These bounds improve upon a bound of 2.45 in the off-line setting or in the randomized on-line setting (Chakrabarti et al. [1996]),,and upon a bound of 3 + epsilon in the deterministic on-line setting (Hall, Schulz, Shmoys and Wein [1996]) Even though the algorithm can be simply stated, the analysis is somewhat intricate. As in Hall et al. [1996], it is based on comparing the weight of the schedule produced to the value of a linear programming (LP) relaxation. However, the algorithm and analysis differ from the approach developed by Hall et al. [1996] in several respects. First, we simultaneously consider two equivalent LP relaxations for the problem: one involving completion time variables, the other preemptive time-indexed variables. The design of the algorithm is essentially based on the time-indexed relaxation; its analysis is based on the completion time relaxation. Moreover, we exploit properties of the optimum solutions to these LP relaxations, and not just of any feasible solution. Another important difference is that we do not relate directly the completion time of job j with its fractional completion time, but only in an amortized sense. Finally, we use randomization to avoid adversarial situations; this last step can also be seen as constructing n different schedules and keeping the best of them.