# Minimizing Maximum (Weighted) Flow-Time on Related and Unrelated Machines

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## Abstract

In this paper we initiate the study of job scheduling on related and unrelated machines so as to minimize the maximum flow time or the maximum weighted flow time (when each job has an associated weight). Previous work for these metrics considered only the setting of parallel machines, while previous work for scheduling on unrelated machines only considered \(L_p, p<\infty \) norms. Our main results are: (1) we give an \(\mathcal {O}({\varepsilon }^{-3})\)-competitive algorithm to minimize maximum weighted flow time on related machines where we assume that the machines of the online algorithm can process \(1+{\varepsilon }\) units of a job in 1 time-unit (\({\varepsilon }\) speed augmentation). (2) For the objective of minimizing maximum flow time on unrelated machines we give a simple \(2/{\varepsilon }\)-competitive algorithm when we augment the speed by \({\varepsilon }\). For *m* machines we show a lower bound of \({\varOmega }(m)\) on the competitive ratio if speed augmentation is not permitted. Our algorithm does not assign jobs to machines as soon as they arrive. To justify this “drawback” we show a lower bound of \({\varOmega }(\log m)\) on the competitive ratio of *immediate dispatch* algorithms. In both these lower bound constructions we use jobs whose processing times are in \(\left\{ 1,\infty \right\} \), and hence they apply to the more restrictive *subset parallel* setting. (3) For the objective of minimizing maximum weighted flow time on unrelated machines we establish a lower bound of \({\varOmega }(\log m)\)-on the competitive ratio of any online algorithm which is permitted to use \(s=\mathcal {O}(1)\) speed machines. In our lower bound construction, job *j* has a processing time of \(p_j\) on a subset of machines and infinity on others and has a weight \(1/p_j\). Hence this lower bound applies to the subset parallel setting for the special case of minimizing maximum stretch.

## Keywords

Scheduling Minimizing flow-time Online algorithms Competitive analysis## References

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