, Volume 77, Issue 2, pp 515–536 | Cite as

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

  • S. Anand
  • Karl Bringmann
  • Tobias FriedrichEmail author
  • Naveen Garg
  • Amit Kumar


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.


Scheduling Minimizing flow-time Online algorithms  Competitive analysis 


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • S. Anand
    • 1
  • Karl Bringmann
    • 2
  • Tobias Friedrich
    • 3
    Email author
  • Naveen Garg
    • 4
  • Amit Kumar
    • 4
  1. 1.Max Planck Institute for InformaticsSaarbrückenGermany
  2. 2.ETH ZürichZurichSwitzerland
  3. 3.Hasso Plattner InstitutePotsdamGermany
  4. 4.Computer Science and EngineeringIndian Institute of Technology DelhiHauz Khas, New DelhiIndia

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