Advertisement

Algorithmica

, Volume 68, Issue 1, pp 62–80 | Cite as

Graph Balancing: A Special Case of Scheduling Unrelated Parallel Machines

  • Tomáš Ebenlendr
  • Marek Krčál
  • Jiří SgallEmail author
Article

Abstract

We design a 1.75-approximation algorithm for a special case of scheduling parallel machines to minimize the makespan, namely the case where each job can be assigned to at most two machines, with the same processing time on either machine. (This is a special case of so-called restricted assignment, where the set of eligible machines can be arbitrary for each job.) This is the first improvement of the approximation ratio 2 of Lenstra, Shmoys, and Tardos (Math. Program. 46:259–271, 1990), to a smaller constant for any special case with unbounded number of machines and unbounded processing times. We conclude by showing integrality gaps of several relaxations of related problems.

Keywords

Feasible Solution Polynomial Time Integral Solution Fractional Solution Unrelated Parallel Machine 
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.

Notes

Acknowledgements

We are grateful to Yossi Azar, Nikhil Bansal and Maxim Sviridenko for discussions concerning this problem. We are grateful to Eiji Miyano for pointing out the work of their group on the same problem. We are also grateful to the anonymous referees for numerous and insightful comments that helped to improve our presentation.

References

  1. 1.
    Asadpour, A., Feige, U., Saberi, A.: Santa Claus meets hypergraph matchings. In: Approximation, Randomization and Combinatorial Optimization: Proc. of the 11th Int. Workshop APPROX 2008 and 12th Int. Workshop RANDOM. Lecture Notes in Comput. Sci., vol. 5171, pp. 10–20. Springer, Berlin (2008) CrossRefGoogle Scholar
  2. 2.
    Asadpour, A., Saberi, A.: An approximation algorithm for max-min fair allocation of indivisible goods. SIAM J. Comput. 39, 2970–2989 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Asahiro, Y., Jansson, J., Miyano, E., Ono, H.: Graph orientation to maximize the minimum weighted outdegree. Int. J. Found. Comput. Sci. 22, 583–601 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Asahiro, Y., Jansson, J., Miyano, E., Ono, H., Zenmyo, K.: Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree. In: Proc. 2nd International Conf. on Algorithmic Aspects in Information and Management (AAIM). Lecture Notes in Comput. Sci., vol. 4508, pp. 167–177. Springer, Berlin (2007) CrossRefGoogle Scholar
  5. 5.
    Asahiro, Y., Jansson, J., Miyano, E., Ono, H., Zenmyo, K.: Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree. J. Comb. Optim. 22, 78–96 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Asahiro, Y., Miyano, E., Ono, H.: Graph classes and the complexity of the graph orientation minimizing the maximum weighted outdegree. Discrete Appl. Math. 159, 498–508 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Asahiro, Y., Miyano, E., Ono, H., Zenmyo, K.: Graph orientation algorithms to minimize the maximum outdegree. Int. J. Found. Comput. Sci. 18, 197–215 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Azar, Y., Epstein, A.: Convex programming for scheduling unrelated parallel machines. In: Proc. 37th Symp. Theory of Computing (STOC), pp. 331–337. ACM, New York (2005) Google Scholar
  9. 9.
    Bansal, N., Sviridenko, M.: The Santa Claus problem. In: Proc. 38th Symp. Theory of Computing (STOC), pp. 31–40. ACM, New York (2006) Google Scholar
  10. 10.
    Bateni, M., Charikar, M., Guruswami, V.: Maxmin allocation via degree lower-bounded arborescences. In: Proc. 41st Symp. Theory of Computing (STOC), pp. 543–552. ACM, New York (2009) Google Scholar
  11. 11.
    Chakrabarty, D., Chuzhoy, J., Khanna, S.: On allocating goods to maximize fairness. In: Proc. 50th Symp. Foundations of Computer Science (FOCS), pp. 107–116. IEEE Press, New York (2009) Google Scholar
  12. 12.
    Ebenlendr, T., Krčál, M., Sgall, J.: Graph balancing: a special case of scheduling unrelated parallel machines. In: Proc. 19th Symp. on Discrete Algorithms (SODA), pp. 483–490. ACM/SIAM, New York (2008) Google Scholar
  13. 13.
    Gairing, M., Monien, B., Woclaw, A.: A faster combinatorial approximation algorithm for scheduling unrelated parallel machines. Theor. Comput. Sci. 380, 87–99 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Haeupler, B., Saha, B., Srinivasan, A.: New constructive aspects of the Lovasz local lemma. In: Proc. 51st Symp. Foundations of Computer Science (FOCS), pp. 397–406. IEEE Press, New York (2010) Google Scholar
  15. 15.
    Hochbaum, D.S., Shmoys, D.: A polynomial approximation scheme for scheduling on uniform processors: Using the dual approximation approach. SIAM J. Comput. 17 (1988) Google Scholar
  16. 16.
    Horowitz, E., Sahni, S.: Exact and approximate algorithms for scheduling nonidentical processors. J. ACM 23, 317–327 (1976) zbMATHMathSciNetGoogle Scholar
  17. 17.
    Jansen, K., Porkolab, L.: Improved approximation schemes for scheduling unrelated parallel machines. Math. Oper. Res. 26, 324–338 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Kumar, V.S.A., Marathe, M.V., Parthasarathy, S., Srinivasan, A.: A unified approach to scheduling on unrelated parallel machines. J. ACM 56, 28 (2009) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Lenstra, J.K., Shmoys, D.B., Tardos, E.: Approximation algorithms for scheduling unrelated parallel machines. Math. Program. 46, 259–271 (1990) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Schuurman, P., Woeginger, G.J.: Polynomial time approximation algorithms for machine scheduling: Ten open problems. J. Sched. 2, 203–213 (1999) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Shchepin, E.V., Vakhania, N.: An optimal rounding gives a better approximation for scheduling unrelated machines. Oper. Res. Lett. 33, 127–133 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Svensson, O.: Santa Claus schedules jobs on unrelated machines. In: Proc. 43rd Symp. Theory of Computing (STOC), pp. 617–626. ACM, New York (2011) Google Scholar
  23. 23.
    Vazirany, V.V.: Approximation Algorithms. Springer, Berlin (2001) Google Scholar
  24. 24.
    Venkateswaran, V.: Minimizing maximum indegree. Discrete Appl. Math. 143, 374–378 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Verschae, J., Wiese, A.: On the configuration-LP for scheduling on unrelated machines. In: Proc. 19th European Symp. on Algorithms (ESA). Lecture Notes in Comput. Sci., vol. 6942, pp. 530–542. Springer, Berlin (2011) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Tomáš Ebenlendr
    • 1
  • Marek Krčál
    • 2
  • Jiří Sgall
    • 3
    Email author
  1. 1.Inst. of Mathematics, AS CRPraha 1Czech Republic
  2. 2.Dept. of Applied MathematicsFaculty of Mathematics and Physics, Charles UniversityPraha 1Czech Republic
  3. 3.Computer Science Inst. of Charles University, Faculty of Mathematics and PhysicsCharles UniversityPraha 1Czech Republic

Personalised recommendations