Generalizing the Kawaguchi-Kyan bound to stochastic parallel machine scheduling

TL;DR

This paper extends the Kawaguchi-Kyan bound to stochastic parallel machine scheduling, providing improved performance ratios for the WSEPT algorithm and matching known bounds in deterministic cases.

Contribution

It generalizes the Kawaguchi-Kyan bound to stochastic settings and refines performance ratios for the WSEPT rule, including exact ratios for fixed machine counts.

Findings

Performance ratio improved to 1+0.5(√2−1)(1+Δ) for stochastic scheduling.

Exact performance ratio of WSPT derived as 0.5(1+√2)−O(1/m^2) for fixed m.

Matching of deterministic bounds in special cases.

Abstract

Minimizing the sum of weighted completion times on $m$ identical parallel machines is one of the most important and classical scheduling problems. For the stochastic variant where processing times of jobs are random variables, M\"ohring, Schulz, and Uetz (1999) presented the first and still best known approximation result achieving, for arbitrarily many machines, performance ratio $1+\frac12(1+\Delta)$, where $\Delta$ is an upper bound on the squared coefficient of variation of the processing times. We prove performance ratio $1+\frac12(\sqrt{2}-1)(1+\Delta)$ for the same underlying algorithm---the Weighted Shortest Expected Processing Time (WSEPT) rule. For the special case of deterministic scheduling (i.e., $\Delta=0$), our bound matches the tight performance ratio $\frac12(1+\sqrt{2})$ of this algorithm (WSPT rule), derived by Kawaguchi and Kyan in a 1986 landmark paper. We present several further improvements for WSEPT's performance ratio, one of them relying on a carefully refined analysis of WSPT yielding, for every fixed number of machines $m$, WSPT's exact performance ratio of order $\frac12(1+\sqrt{2})-O(1/m^2)$.