Scheduling and fixed-parameter tractability

Abstract

Fixed-parameter tractability analysis and scheduling are two core domains of combinatorial optimization which led to deep understanding of many important algorithmic questions. However, even though fixed-parameter algorithms are appealing for many reasons, no such algorithms are known for many fundamental scheduling problems. In this paper we present the first fixed-parameter algorithms for classical scheduling problems such as makespan minimization, scheduling with job-dependent cost functions—one important example being weighted flow time—and scheduling with rejection. To this end, we identify crucial parameters that determine the problems’ complexity. In particular, we manage to cope with the problem complexity stemming from numeric input values, such as job processing times, which is usually a core bottleneck in the design of fixed-parameter algorithms. We complement our algorithms with \(\mathsf {W[1]}\)-hardness results showing that for smaller sets of parameters the respective problems do not allow fixed-parameter algorithms. In particular, our positive and negative results for scheduling with rejection explore a research direction proposed by Dániel Marx.

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Acknowledgments

We thank the reviewers of IPCO 2014 and Mathematical Programming for their remarks on improving the presentation of the results. We further thank an anonymous reviewer of an earlier version for suggestions how to improve the algorithms in Sect. 4 and to prove Theorem 9, as well as Akiyoshi Shioura for helpful discussions about convexity. Finally, we would like to thank the anonymous reviewers for pointing out the IP formulation in Sect. 2.2 which is simpler than our original formulation.

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Correspondence to Andreas Wiese.

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A preliminary version of this article appeared in the proceedings of IPCO 2014 [34].

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Mnich, M., Wiese, A. Scheduling and fixed-parameter tractability. Math. Program. 154, 533–562 (2015). https://doi.org/10.1007/s10107-014-0830-9

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Keywords

  • Scheduling
  • Fixed-parameter tractability
  • Integer linear programming

Mathematics Subject Classification

  • 68W05
  • 90B35