Structural Parameters for Scheduling with Assignment Restrictions

  • Klaus Jansen
  • Marten Maack
  • Roberto Solis-Oba
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10236)


We consider scheduling on identical and unrelated parallel machines with job assignment restrictions. These problems are NP-hard and they do not admit polynomial time approximation algorithms with approximation ratios smaller than 1.5 unless P = NP. However, if we impose limitations on the set of machines that can process a job, the problem sometimes becomes easier in the sense that algorithms with approximation ratios better than 1.5 exist. We introduce three graphs, based on the assignment restrictions and study the computational complexity of the scheduling problem with respect to structural properties of these graphs, in particular their tree- and rankwidth. We identify cases that admit polynomial time approximation schemes or FPT algorithms, generalizing and extending previous results in this area.



The Rounding Lemma in the presented form was formulated by Lars Rohwedder and Kevin Prohn as part of a student project.


  1. 1.
    Szeider, S.: On fixed-parameter tractable parameterizations of SAT. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 188–202. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-24605-3_15 CrossRefGoogle Scholar
  2. 2.
    Samer, M., Szeider, S.: Constraint satisfaction with bounded treewidth revisited. J. Comput. Syst. Sci. 76(2), 103–114 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Lenstra, J.K., Shmoys, D.B., Tardos, É.: Approximation algorithms for scheduling unrelated parallel machines. Math. Program. 46(1–3), 259–271 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Horowitz, E., Sahni, S.: Exact and approximate algorithms for scheduling nonidentical processors. J. ACM (JACM) 23(2), 317–327 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ebenlendr, T., Krčál, M., Sgall, J.: Graph balancing: a special case of scheduling unrelated parallel machines. Algorithmica 68(1), 62–80 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lee, K., Leung, J.Y.T., Pinedo, M.L.: A note on graph balancing problems with restrictions. Inf. Process. Lett. 110(1), 24–29 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Asahiro, Y., Miyano, E., Ono, H.: Graph classes and the complexity of the graph orientation minimizing the maximum weighted outdegree. Discret. Appl. Math. 159(7), 498–508 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ou, J., Leung, J.Y.T., Li, C.L.: Scheduling parallel machines with inclusive processing set restrictions. Naval Res. Logist. (NRL) 55(4), 328–338 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Epstein, L., Levin, A.: Scheduling with processing set restrictions: PTAS results for several variants. Int. J. Prod. Econ. 133(2), 586–595 (2011)CrossRefGoogle Scholar
  10. 10.
    Muratore, G., Schwarz, U.M., Woeginger, G.J.: Parallel machine scheduling with nested job assignment restrictions. Oper. Res. Lett. 38(1), 47–50 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mnich, M., Wiese, A.: Scheduling and fixed-parameter tractability. Math. Program. 154(1–2), 533–562 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Knop, D., Kouteckỳ, M.: Scheduling meets n-fold integer programming. arXiv preprint arXiv:1603.02611 [cs.DS] (2016)
  13. 13.
    Szeider, S.: Not so easy problems for tree decomposable graphs. In: International Conference on Discrete Mathematics (2008)Google Scholar
  14. 14.
    Giakoumakis, V., Vanherpe, J.M.: Linear time recognition of weak bisplit graphs. Electron. Notes Discret. Math. 5, 138–141 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jansen, K., Maack, M., Solis-Oba, R.: Structural parameters for scheduling with assignment restrictions. arXiv preprint arXiv:1701.07242 [cs.DS] (2017)
  16. 16.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theoret. Comput. Sci. 209(1), 1–45 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Algebraic Discret. Methods 8(2), 277–284 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hlinenỳ, P., Oum, S.I.: Finding branch-decompositions and rank-decompositions. SIAM J. Comput. 38(3), 1012–1032 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kolaitis, P.G., Vardi, M.Y.: Conjunctive-query containment and constraint satisfaction. J. Comput. Syst. Sci. 61, 302–332 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Christian-Albrechts-Universität zu KielKielGermany
  2. 2.Western UniversityLondonCanada

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