Precedence-Constrained Scheduling Problems Parameterized by Partial Order Width

  • René van BevernEmail author
  • Robert Bredereck
  • Laurent Bulteau
  • Christian Komusiewicz
  • Nimrod Talmon
  • Gerhard J. Woeginger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9869)


Negatively answering a question posed by Mnich and Wiese (Math. Program. 154(1–2):533–562), we show that \(\hbox {P2}|\hbox {prec}, p_{j}{\in }\{1,2\}|C_{\max }\), the problem of finding a non-preemptive minimum-makespan schedule for precedence-constrained jobs of lengths 1 and 2 on two parallel identical machines, is W[2]-hard parameterized by the width of the partial order giving the precedence constraints. To this end, we show that Shuffle Product, the problem of deciding whether a given word can be obtained by interleaving the letters of \(k\) other given words, is W[2]-hard parameterized by \(k\), thus additionally answering a question posed by Rizzi and Vialette (CSR 2013). Finally, refining a geometric algorithm due to Servakh (Diskretn. Anal. Issled. Oper. 7(1):75–82), we show that the more general Resource-Constrained Project Scheduling problem is fixed-parameter tractable parameterized by the partial order width combined with the maximum allowed difference between the earliest possible and factual starting time of a job.


Resource-constrained project scheduling Parallel identical machines Makespan minimization Parameterized complexity Shuffle product 



The authors are thankful to Sergey Sevastyanov for pointing out the work of Akers [1] and Servakh [14]. This research was initiated at the annual research retreat of the algorithms and complexity group of TU Berlin, April 3–9, 2016, Krölpa, Germany.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • René van Bevern
    • 1
    Email author
  • Robert Bredereck
    • 2
  • Laurent Bulteau
    • 3
  • Christian Komusiewicz
    • 4
  • Nimrod Talmon
    • 5
  • Gerhard J. Woeginger
    • 6
  1. 1.Novosibirsk State UniversityNovosibirskRussian Federation
  2. 2.TU BerlinBerlinGermany
  3. 3.Université Paris-Est Marne-la-ValléeChamps-sur-MarneFrance
  4. 4.Friedrich-Schiller-Universität JenaJenaGermany
  5. 5.Weizmann Institute of ScienceRehovotIsrael
  6. 6.TU EindhovenEindhovenThe Netherlands

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