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Algorithmica

pp 1–19 | Cite as

Non-clairvoyantly Scheduling to Minimize Convex Functions

  • Kyle Fox
  • Sungjin Im
  • Janardhan KulkarniEmail author
  • Benjamin Moseley
Article
  • 10 Downloads

Abstract

The paper considers scheduling jobs online to minimize the objective \(\sum _{i \in [n]}w_ig(C_i-r_i)\), where \(w_i\) is the weight of job i, \(r_i\) is its release time, \(C_i\) is its completion time and g is any non-decreasing convex function. It is known that the clairvoyant algorithm Highest-Density-First (HDF) is \((2+\epsilon )\)-speed O(1)-competitive for this objective on a single machine for any fixed \( 0< \epsilon < 1\) (Im et al., in: ACM-SIAM symposium on discrete algorithms, pp 1254–1265, 2012). In this paper, we give the first non-trivial results for this problem when g is a non-decreasing convex function and the algorithm must be non-clairvoyant. More specifically, our results include:
  • A \((2+\epsilon )\)-speed O(1)-competitive non-clairovyant algorithm on a single machine for all non-decreasing convex g, matching the performance of HDF for any fixed \( 0< \epsilon < 1\).

  • A \((3+\epsilon )\)-speed O(1)-competitive non-clairovyant algorithm on multiple identical machines for all non-decreasing convex g for any fixed \( 0< \epsilon < 1\).

The paper gives the first non-trivial upper-bound on multiple machines even if the algorithm is allowed to be clairvoyant. All performance guarantees above hold for all non-decreasing convex functions gsimultaneously. The positive results are supplemented by almost matching lower bounds. We show that any algorithm that is oblivious to g is not O(1)-competitive with speed augmentation less than 2 on a single machine. Further, any non-clairvoyent algorithm that knows the function g cannot be O(1)-competitive with speed augmentation less than \(\sqrt{2}\) on a single machine or  \((2-\frac{1}{m})\) on m identical machines.

Keywords

Online algorithms Scheduling theory Competitive analysis 

Notes

Acknowledgements

The authors would like to thank the anonymous reviewers of previous versions of this paper for their helpful, and sometimes quite detailed, comments and suggestions.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Kyle Fox
    • 1
  • Sungjin Im
    • 2
  • Janardhan Kulkarni
    • 3
    Email author
  • Benjamin Moseley
    • 4
  1. 1.University of Texas at DallasRichardsonUSA
  2. 2.University of California-MercedMercedUSA
  3. 3.Microsoft ResearchRedmondUSA
  4. 4.Carnegie Mellon UniversityPittsburghUSA

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