Approximation in stochastic scheduling: The power of LP-based priority policies

Rolf H. Möhring, Andreas S. Schulz, and Marc Uetz

Extended abstract in: Randomization, Approximation, and Combinatorial Optimization, Dorit Hochbaum, Klaus Jansen, Jose D. P. Rolim, Alistair Sinclair (eds.), Lecture Notes in Computer Science 1671, Springer, Berlin, 1999, pages 144-155, Journal version: Journal of the ACM 46(6), 1999, pages 924-942.

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Stochastic scheduling, Approximation, Worst-case performance, Priority policy, LP-relaxation, WSEPT rule, Asymptotic optimality

We consider the problem to minimize the total weighted completion time of a set of jobs with individual release dates which have to be scheduled on identical parallel machines. Job processing times are not known in advance, they are realized on-line according to given probability distributions. The aim is to find a scheduling policy that minimizes the objective in expectation. Motivated by the success of LP-based approaches to deterministic scheduling, we present a polyhedral relaxation of the performance space of stochastic parallel machine scheduling. This relaxation extends earlier relaxations that have been used, among others, by Hall, Schulz, Shmoys, and Wein (1997) in the deterministic setting. We then derive constant performance guarantees for priority policies which are guided by optimum LP solutions, and thereby generalize previous results from deterministic scheduling. In the absence of release dates, the LP-based analysis also yields an additive performance guarantee for the WSEPT rule which implies both a worst-case performance ratio and a result on its asymptotic optimality, thus complementing previous work by Weiss (1990). The corresponding LP lower bound generalizes a previous lower bound from deterministic scheduling due to Eastman, Even, and Isaacs (1964), and exhibits a relation between parallel machine problems and corresponding problems with only one fast single machine. Finally, we show that all employed LPs can be solved in polynomial time by purely combinatorial algorithms.

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