Approximation in Stochastic Scheduling: the Power of Lp-based Priority Rules Approximation in Stochastic Scheduling: the Power of Lp-based Priority Rules

We address the problem of minimizing the total weighted completion time in stochastic machine scheduling. A set of jobs has to be processed on identical parallel machines, job processing times are drawn from some known probability distribution, and the aim is to find a scheduling strategy that minimizes the objective in expectation. The overall driving question is: What is the performance of simple scheduling strategies like, e.g., the WSEPT rule in this stochastic setting? An answer to this question is important for mainly two reasons: On the one hand priority rules are widely used in practice, yet their use is theoretically justified only rarely and under rather restrictive assumptions. Our main contribution is the first non-trivial polyhedral relaxation of the set of all expected completion time vectors which correspond to feasible scheduling strategies. This relaxation is used to derive provably good priority rules in a general stochastic setting. We thereby generalize previous results for deterministic scheduling for both the single and the parallel machine case. More precisely, we derive constant-factor performance guarantees for priority rules which are guided by optimum LP solutions. It is an important feature that both the LP relaxations as well as the corresponding LP-based priority rules only rely on the expected job processing times. That is, although they do not require further information on the distribution they are nevertheless close to an optimal strategy which has full information. For instance, we show that the WSEPT rule is a (2 1 m)-approximation whenever the processing times are drawn from NBUE distributions. Most of our results are based on relaxations of parallel machine problems which correspond to ‘fast’ single machine problems. For single machine scheduling, this leads to an exact polyhedral description of the convex hull of expected completion time vectors.