Generalizing the Kawaguchi-Kyan bound to stochastic parallel machine scheduling
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Minimizing the sum of weighted completion times on m identical parallel machines is one of the most important classical scheduling problems. For the stochastic variant where processing times of jobs are random variables, in 1999, Möhring, Schulz, and Uetz presented the first and still best known approximation result, achieving, for arbitrarily many machines, performance guarantee 1 + 1 2 · (1+Δ), where Δ is an upper bound on the squared coefficient of variation of the processing times. We prove performance guarantees 1 + 1 2 (√(2) - 1) · (1+Δ) for arbitrary Δ and 1 + 1 6 · (1+Δ) for Δ> 1 for the same underlying algorithm--the Weighted Shortest Expected Processing Time (WSEPT) rule. For the special case of deterministic scheduling (i.e. Δ = 0), our bound matches the tight performance ratio 1 2 (1+√(2)) of this algorithm (WSPT rule) by Kawaguchi and Kyan from 1986.
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