Document Open Access Logo

Min-Sum Scheduling Under Precedence Constraints

Authors Andreas S. Schulz, José Verschae

PDF
Thumbnail PDF

File

LIPIcs.ESA.2016.74.pdf
  • Filesize: 0.49 MB
  • 13 pages

Document Identifiers

Author Details

Andreas S. Schulz
José Verschae

Cite AsGet BibTex

Andreas S. Schulz and José Verschae. Min-Sum Scheduling Under Precedence Constraints. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 74:1-74:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ESA.2016.74

Abstract

In many scheduling situations, it is important to consider non-linear functions of job completions times in the objective. This was already recognized by Smith (1956). Recently, the theory community has begun a thorough study of the resulting problems, mostly on single-machine instances for which all permutations of jobs are feasible. However, a typical feature of many scheduling problems is that some jobs can only be processed after others. In this paper, we give the first approximation algorithms for min-sum scheduling with (nonnegative, non-decreasing) non-linear functions and general precedence constraints. In particular, for 1|prec|sum w_j f(C_j), we propose a polynomial-time universal algorithm that performs well for all functions f simultaneously. Its approximation guarantee is 2 for all concave functions, at worst. We also provide a (non-universal) polynomial-time algorithm for the more general case 1|prec|sum f_j(C_j). The performance guarantee is no worse than 2+epsilon for all concave functions. Our results match the best bounds known for the case of linear functions, a widely studied problem, and considerably extend the results for minimizing sum w_jf(C_j) without precedence constraints.
Keywords
  • scheduling
  • approximation algorithms
  • linear programming relaxations
  • precedence constraints

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. N. Bansal and S. Khot. Optimal long code test with one free bit. In Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009), pages 453-462. IEEE, 2009. Google Scholar
  2. N. Bansal and K. Pruhs. The geometry of scheduling. In Proceedings of the 51th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pages 407-414. IEEE, 2010. Google Scholar
  3. N. Bansal and K. Pruhs. The geometry of scheduling. SIAM Journal on Computing, 43:1684-1698, 2014. Google Scholar
  4. C. Chekuri and R. Motwani. Precedence constrained scheduling to minimize sum of weighted completion times on a single machine. Discrete Applied Mathematics, 98:29-38, 1999. Google Scholar
  5. M. Cheung and D. Shmoys. A primal-dual approximation algorithm for min-sum single-machine scheduling problems. In Proceedings of the 14th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2011), volume 6845 of Lecture Notes in Computer Science, pages 135-146. Springer, 2011. Google Scholar
  6. F. A. Chudak and D. S. Hochbaum. A half-integral linear programming relaxation for scheduling precedence-constrained jobs on a single machine. Operations Research Letters, 25:199-204, 1999. Google Scholar
  7. J. R. Correa and A. S. Schulz. Single-machine scheduling with precedence constraints. Mathematics of Operations Research, 30:1005-1021, 2005. Google Scholar
  8. L. Epstein, A. Levin, A. Marchetti-Spaccamela, N. Megow, J. Mestre, M. Skutella, and L. Stougie. Universal sequencing on a single machine. SIAM Journal on Computing, 41:565-586, 2012. Google Scholar
  9. M. X. Goemans. Improved approximation algorithms for scheduling with release dates. In Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1997), pages 591-598, 1997. Google Scholar
  10. R. A. Gordon. The Integrals of Lebesgue, Denjoy, Perron, and Henstock. American Mathematical Society, 1994. Google Scholar
  11. R. L. Graham, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan. Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics, 5:287-326, 1979. Google Scholar
  12. L. A. Hall, A. S. Schulz, D. B. Shmoys, and J. Wein. Scheduling to minimize average completion time: Off-line and on-line approximation algorithms. Mathematics of Operations Research, 22:513-544, 1997. Google Scholar
  13. W. Höhn and T. Jacobs. On the performance of Smith’s rule in single-machine scheduling with nonlinear cost. ACM Transactions on Algorithms, 11, 2015. Google Scholar
  14. S. Im, B. Moseley, and K. Pruhs. Online scheduling with general cost function. In Proceedings of the 23rd Annual Symposium on Discrete Algorithms (SODA 2012), pages 1254-1265, 2012. Google Scholar
  15. S. G. Kolliopoulos and G. Steiner. Partially ordered knapsack and applications to scheduling. Discrete Applied Mathematics, 155:889-897, 2007. Google Scholar
  16. F. Margot, M. Queyranne, and Y. Wang. Decompositions, network flows, and a precedence constrained single-machine scheduling problem. Operations Research, 51:981-992, 2003. Google Scholar
  17. N. Megow and J. Verschae. Dual techniques for scheduling on a machine with varying speed. In Automata, Languages, and Programming (ICALP 2013), volume 7965 of Lecture Notes in Computer Science, pages 745-756, 2013. Google Scholar
  18. J. Mestre and J. Verschae. A 4-approximation for scheduling on a single machine with general cost function. arXiv:1403.0298, 2014. Google Scholar
  19. A. S. Schulz. Scheduling to minimize total weighted completion time: Performance guarantees of LP-based heuristics and lower bounds. In Integer Programming and Combinatorial Optimization (proceedings of IPCO V), volume 1084 of Lecture Notes in Computer Science, pages 301-315, 1996. Google Scholar
  20. A. S. Schulz and M. Skutella. Random-based scheduling: New approximations and LP lower bounds. In Randomization and Approximation Techniques in Computer Science, volume 1269 of Lecture Notes in Computer Science, pages 119-133, 1997. Google Scholar
  21. J. B. Sidney. Decomposition algorithms for single machine scheduling with precedence relations and deferral costs. Operations Research, 23:283-298, 1975. Google Scholar
  22. M. Skutella. List scheduling in order of α-points on a single machine. In E. Bampis, K. Jansen, and C. Kenyon, editors, Efficient Approximation and Online Algorithms: Recent Progress on Classical Combinatorial Optimization Problems and New Applications, volume 3484 of Lecture Notes in Computer Science, pages 250-291. Springer, 2006. Google Scholar
  23. S. Stiller and A. Wiese. Increasing speed scheduling and flow scheduling. In Proceedings of the 21st Symposium on Algorithms and Computation (ISAAC 2010), volume 6507 of Lecture Notes in Computer Science, pages 279-290, 2010. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact