Skip to main content
Springer Nature Link
Log in

Scheduling with Outliers

  • Conference paper
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2009, RANDOM 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5687))

  • 1909 Accesses

Abstract

In classical scheduling problems, we are given jobs and machines, and have to schedule all the jobs to minimize some objective function. What if each job has a specified profit, and we are no longer required to process all jobs? Instead, we can schedule any subset of jobs whose total profit is at least a (hard) target profit requirement, while still trying to approximately minimize the objective function.

We refer to this class of problems as scheduling with outliers. This model was initiated by Charikar and Khuller (SODA ’06) for minimum max-response time in broadcast scheduling. In this paper, we consider three other well-studied scheduling objectives: the generalized assignment problem, average weighted completion time, and average flow time, for which LP-based approximation algorithms are provided. Our main results are:

  • For the minimum average flow time problem on identical machines, we give an LP-based logarithmic approximation algorithm for the unit profits case, and complement this result by presenting a matching integrality gap.

  • For the average weighted completion time problem on unrelated machines, we give a constant-factor approximation. The algorithm is based on randomized rounding of the time-indexed LP relaxation strengthened by knapsack-cover inequalities.

  • For the generalized assignment problem with outliers, we outline a simple reduction to GAP without outliers to obtain an algorithm whose makespan is within 3 times the optimum makespan, and whose cost is at most (1 + ε) times the optimal cost.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Mayr, E.W., Prömel, H.J., Steger, A. (eds.): Dagstuhl Seminar 1997. LNCS, vol. 1367. Springer, Heidelberg (1998)

    MATH  Google Scholar 

  2. Bansal, N., Blum, A., Chawla, S., Dhamdhere, K.: Scheduling for flow-time with admission control. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 43–54. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  3. Bartal, Y., Leonardi, S., Marchetti-Spaccamela, A., Sgall, J., Stougie, L.: Multiprocessor scheduling with rejection. In: SODA 1996, pp. 95–103 (1996)

    Google Scholar 

  4. Carr, R.D., Fleischer, L., Leung, V.J., Phillips, C.A.: Strengthening integrality gaps for capacitated network design and covering problems. In: SODA 2000, pp. 106–115 (2000)

    Google Scholar 

  5. Charikar, M., Khuller, S.: A robust maximum completion time measure for scheduling. In: SODA 2006, pp. 324–333 (2006)

    Google Scholar 

  6. Charikar, M., Khuller, S., Mount, D.M., Narasimhan, G.: Algorithms for facility location problems with outliers. In: SODA 2001, pp. 642–651 (2001)

    Google Scholar 

  7. Chen, K.: A constant factor approximation algorithm for k-median clustering with outliers. In: SODA 2008, pp. 826–835 (2008)

    Google Scholar 

  8. Chudak, F.A., Roughgarden, T., Williamson, D.P.: Approximate k-msts and k-steiner trees via the primal-dual method and lagrangean relaxation. Mathematical Programming 100(2), 411–421 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Engels, D.W., Karger, D.R., Kolliopoulos, S.G., Sengupta, S., Uma, R.N., Wein, J.: Techniques for scheduling with rejection. In: Bilardi, G., Pietracaprina, A., Italiano, G.F., Pucci, G. (eds.) ESA 1998. LNCS, vol. 1461, pp. 175–191. Springer, Heidelberg (1998)

    Google Scholar 

  10. Epstein, L., Noga, J., Woeginger, G.J.: On-line scheduling of unit time jobs with rejection: minimizing the total completion time. Operations Research Letters 30(6), 415–420 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Garg, N.: Saving an epsilon: a 2-approximation for the k-MST problem in graphs. In: STOC 2005, pp. 396–402 (2005)

    Google Scholar 

  12. Garg, N., Kumar, A.: Better algorithms for minimizing average flow-time on related machines. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 181–190. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  13. Garg, N., Kumar, A.: Minimizing average flow-time: Upper and lower bounds. In: FOCS 2007, pp. 603–613 (2007)

    Google Scholar 

  14. Golovin, D., Nagarajan, V., Singh, M.: Approximating the k-multicut problem. In: SODA 2006, pp. 621–630 (2006)

    Google Scholar 

  15. Guha, S., Munagala, K.: Model-driven optimization using adaptive probes. In: SODA 2007, pp. 308–317 (2007)

    Google Scholar 

  16. Hall, L.A., Shmoys, D.B., Wein, J.: Scheduling to minimize average completion time: off-line and on-line algorithms. In: SODA 1996, pp. 142–151 (1996)

    Google Scholar 

  17. Hoogeveen, H., Skutella, M., Woeginger, G.J.: Preemptive scheduling with rejection. Mathematical Programming, Ser. B 94(2-3), 361–374 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation. Journal of the ACM 48(2), 274–296 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Könemann, J., Parekh, O., Segev, D.: A unified approach to approximating partial covering problems. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 468–479. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  20. Lenstra, J.K., Shmoys, D.B., Tardos, É.: Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming 46, 259–271 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Leonardi, S., Raz, D.: Approximating total flow time on parallel machines. In: STOC 1997, pp. 110–119 (1997)

    Google Scholar 

  22. Levin, A., Segev, D.: Partial multicuts in trees. Theoretical Computer Science 369(1-3), 384–395 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mestre, J.: A primal-dual approximation algorithm for partial vertex cover: making educated guesses. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 182–191. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  24. Mestre, J.: Lagrangian relaxation and partial cover (extended abstract). In: STACS 2008, pp. 539–550 (2008)

    Google Scholar 

  25. Schulz, A.S., Skutella, M.: Scheduling unrelated machines by randomized rounding. SIAM Journal on Discrete Mathematics 15(4), 450–469 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Seiden, S.S.: Preemptive multiprocessor scheduling with rejection. Theoretical Computer Science 262(1), 437–458 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shmoys, D.B., Tardos, É.: An approximation algorithm for the generalized assignment problem. Mathematical Programming 62, 461–474 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  28. Skutella, M.: Convex quadratic and semidefinite programming relaxations in scheduling. Journal of the ACM 48(2), 206–242 (2001)

    Article  MathSciNet  Google Scholar 

  29. Wolsey, L.A.: Faces for a linear inequality in 0-1 variables. Mathematical Programming 8, 165–178 (1975)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Carnegie Mellon University, USA

    Anupam Gupta & Ravishankar Krishnaswamy

  2. Department of Computer Science and Engineering, Indian Institute of Technology, New Delhi, India, 110016

    Amit Kumar

  3. Sloan School of Management, Massachusetts Institute of Technology, USA

    Danny Segev

Authors
  1. Anupam Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ravishankar Krishnaswamy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Amit Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Danny Segev
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. of Applied Math and Computer Science, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Irit Dinur

  2. Institute for Computer Science and Applied Mathematics, University of Kiel, Olshausenstr. 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Technion, Computer Science Department, 32000, Haifa, Israel

    Joseph Naor

  4. University of Geneva, Centre Universitaire d’Informatique, Battelle Bat. A, 7 rte de Drize, 1227, Carouge, Switzerland

    José Rolim

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gupta, A., Krishnaswamy, R., Kumar, A., Segev, D. (2009). Scheduling with Outliers. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-03685-9_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03684-2

  • Online ISBN: 978-3-642-03685-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics