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How Unsplittable-Flow-Covering Helps Scheduling with Job-Dependent Cost Functions

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8572)

Abstract

Generalizing many well-known and natural scheduling problems, scheduling with job-specific cost functions has gained a lot of attention recently. In this setting, each job incurs a cost depending on its completion time, given by a private cost function, and one seeks to schedule the jobs to minimize the total sum of these costs. The framework captures many important scheduling objectives such as weighted flow time or weighted tardiness. Still, the general case as well as the mentioned special cases are far from being very well understood yet, even for only one machine. Aiming for better general understanding of this problem, in this paper we focus on the case of uniform job release dates on one machine for which the state of the art is a 4-approximation algorithm. This is true even for a special case that is equivalent to the covering version of the well-studied and prominent unsplittable flow on a path problem, which is interesting in its own right. For that covering problem, we present a quasi-polynomial time (1 + ε)-approximation algorithm that yields an (e + ε)-approximation for the above scheduling problem. Moreover, for the latter we devise the best possible resource augmentation result regarding speed: a polynomial time algorithm which computes a solution with optimal cost at 1 + ε speedup. Finally, we present an elegant QPTAS for the special case where the cost functions of the jobs fall into at most logn many classes. This algorithm allows the jobs even to have up to logn many distinct release dates. All proposed quasi-polynomial time algorithms require the input data to be quasi-polynomially bounded.

Keywords

  • Cost Function
  • Approximation Algorithm
  • Completion Time
  • Release Date
  • Polynomial Time Algorithm

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Funded by the Go8-DAAD joint research cooperation scheme.

A full version of the paper can be found at http://arxiv.org/abs/1403.1376 .

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References

  1. Afrati, F., Bampis, E., Chekuri, C., Karger, D., Kenyon, C., Khanna, S., Milis, I., Queyranne, M., Skutella, M., Stein, C., Sviridenko, M.: Approximation schemes for minimizing average weighted completion time with release dates. In: Proceedings of FOCS 1999, pp. 32–44 (1999)

    Google Scholar 

  2. Anagnostopoulos, A., Grandoni, F., Leonardi, S., Wiese, A.: A mazing 2+ε approximation for unsplittable flow on a path. In: Proceedings of SODA 2014, pp. 26–41 (2014)

    Google Scholar 

  3. Bansal, N., Chakrabarti, A., Epstein, A., Schieber, B.: A quasi-PTAS for unsplittable flow on line graphs. In: Proceedings of STOC 2006, pp. 721–729 (2006)

    Google Scholar 

  4. Bansal, N., Dhamdhere, K.: Minimizing weighted flow time. ACM T. Alg. 3(4), article 39 (2007)

    Google Scholar 

  5. Bansal, N., Friggstad, Z., Khandekar, R., Salavatipour, R.: A logarithmic approximation for unsplittable flow on line graphs. In: Proceedings of SODA 2009, pp. 702–709 (2009)

    Google Scholar 

  6. Bansal, N., Pruhs, K.: The geometry of scheduling. In: Proceedings of FOCS 2010, pp. 407–414 (2010), See also http://www.win.tue.nl/~nikhil/pubs/wflow-journ3.pdf

  7. Bansal, N., Pruhs, K.: Weighted geometric set multi-cover via quasi-uniform sampling. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 145–156. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  8. Bansal, N., Verschae, J.: Personal communication

    Google Scholar 

  9. Bar-Noy, A., Bar-Yehuda, R., Freund, A., Naor, J., Schieber, B.: A unified approach to approximating resource allocation and scheduling. J. ACM 48(5), 1069–1090 (2001)

    CrossRef  MathSciNet  Google Scholar 

  10. Bar-Yehuda, R., Rawitz, D.: On the equivalence between the primal-dual schema and the local ratio technique. SIAM J. Discrete Math. 19(3), 762–797 (2005)

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. Bonsma, P., Schulz, J., Wiese, A.: A constant factor approximation algorithm for unsplittable flow on paths. In: Proceedings of FOCS 2011, pp. 47–56 (2011)

    Google Scholar 

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

    Google Scholar 

  13. Chakaravarthy, V.T., Kumar, A., Roy, S., Sabharwal, Y.: Resource allocation for covering time varying demands. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 543–554. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  14. Chakrabarti, A., Chekuri, C., Gupta, A., Kumar, A.: Approximation algorithms for the unsplittable flow problem. In: Jansen, K., Leonardi, S., Vazirani, V.V. (eds.) APPROX 2002. LNCS, vol. 2462, pp. 51–66. Springer, Heidelberg (2002)

    CrossRef  Google Scholar 

  15. Chekuri, C., Khanna, S.: Approximation schemes for preemptive weighted flow time. In: Proceedings of STOC 2002, pp. 297–305 (2002)

    Google Scholar 

  16. Chekuri, C., Khanna, S., Zhu, A.: Algorithms for minimizing weighted flow time. In: Proceedings of STOC 2001, pp. 84–93 (2001)

    Google Scholar 

  17. Chekuri, C., Mydlarz, M., Shepherd, F.: Multicommodity demand flow in a tree and packing integer programs. ACM T. Alg. 3(3), article 27 (2007)

    Google Scholar 

  18. Cheung, M., Shmoys, D.B.: A primal-dual approximation algorithm for min-sum single-machine scheduling problems. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds.) RANDOM 2011 and APPROX 2011. LNCS, vol. 6845, pp. 135–146. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  19. Kao, M.-Y., Reif, J.H., Tate, S.R.: Searching in an unknown environment: An optimal randomized algorithm for the cow-path problem. Inform. Comput. 131(1), 63–79 (1996)

    CrossRef  MATH  MathSciNet  Google Scholar 

  20. Lawler, E.L.: A “pseudopolynomial” algorithm for sequencing jobs to minimize total tardiness. Ann. Discrete Math. 1, 331–342 (1977)

    CrossRef  MathSciNet  Google Scholar 

  21. Megow, N., Verschae, J.: Dual techniques for scheduling on a machine with varying speed. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 745–756. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  22. Mestre, J., Verschae, J.: A 4-approximation for scheduling on a single machine with general cost function, http://arxiv.org/abs/1403.0298

  23. Shmoys, D.B., Tardos, É.: An approximation algorithm for the generalized assignment problem. Math. Program. 62(1-3), 461–474 (1993)

    CrossRef  MATH  MathSciNet  Google Scholar 

  24. Sviridenko, M., Wiese, A.: Approximating the configuration-LP for minimizing weighted sum of completion times on unrelated machines. In: Goemans, M., Correa, J. (eds.) IPCO 2013. LNCS, vol. 7801, pp. 387–398. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

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Höhn, W., Mestre, J., Wiese, A. (2014). How Unsplittable-Flow-Covering Helps Scheduling with Job-Dependent Cost Functions. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43948-7_52

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  • DOI: https://doi.org/10.1007/978-3-662-43948-7_52

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