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Interval scheduling and colorful independent sets

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Abstract

Numerous applications in scheduling, such as resource allocation or steel manufacturing, can be modeled using the NP-hard Independent Set problem (given an undirected graph and an integer \(k\), find a set of at least \(k\) pairwise non-adjacent vertices). Here, one encounters special graph classes like 2-union graphs (edge-wise unions of two interval graphs) and strip graphs (edge-wise unions of an interval graph and a cluster graph), on which Independent Set remains \(\mathrm{NP}\)-hard but admits constant ratio approximations in polynomial time. We study the parameterized complexity of Independent Set on 2-union graphs and on subclasses like strip graphs. Our investigations significantly benefit from a new structural “compactness” parameter of interval graphs and novel problem formulations using vertex-colored interval graphs. Our main contributions are as follows:

  1. 1.

    We show a complexity dichotomy: restricted to graph classes closed under induced subgraphs and disjoint unions, Independent Set is polynomial-time solvable if both input interval graphs are cluster graphs, and is \(\mathrm{NP}\)-hard otherwise.

  2. 2.

    We chart the possibilities and limits of effective polynomial-time preprocessing (also known as kernelization).

  3. 3.

    We extend Halldórsson and Karlsson (2006)’s fixed-parameter algorithm for Independent Set on strip graphs parameterized by the structural parameter “maximum number of live jobs” to show that the problem (also known as Job Interval Selection) is fixed-parameter tractable with respect to the parameter \(k\) and generalize their algorithm from strip graphs to 2-union graphs. Preliminary experiments with random data indicate that Job Interval Selection with up to 15 jobs and \(5\cdot 10^5\) intervals can be solved optimally in less than 5 min.

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Notes

  1. The Exponential Time Hypothesis basically states that there is no \(2^{o(n)}\)-time algorithm for \(n\)-variable 3-SAT (Impagliazzo et al. 2001; Lokshtanov et al. 2011).

  2. http://fpt.akt.tu-berlin.de/cis/.

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Acknowledgments

We thank Michael Dom and Hannes Moser for discussions on coil coating, which initiated our investigations on 2-Union Independent Set , as well as Wiebke Höhn for providing details regarding the application of 2-Union Independent Set  in steel manufacturing. René van Bevern was supported by the Deutsche Forschungsgemeinschaft (DFG), project DAPA, NI 369/12. Part of the work was done while being supported by DFG project AREG, NI 369/9. Mathias Weller was supported by the DFG, project DARE, NI 369/11.

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Authors and Affiliations

  1. Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Berlin, Germany

    René van Bevern & Rolf Niedermeier

  2. Cluster of Excellence Multimodal Computing and Interaction, Saarbrücken, Germany

    Matthias Mnich

  3. LIRMM, University Montpellier II, Montpellier, France

    Mathias Weller

Authors
  1. René van Bevern
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  2. Matthias Mnich
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  3. Rolf Niedermeier
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  4. Mathias Weller
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Corresponding author

Correspondence to René van Bevern.

Additional information

A preliminary version of this article appeared in the proceedings of the 23rd International Symposium on Algorithms and Computation (ISAAC 2012), volume 7676 in Lecture Notes in Computer Science, pp. 247–256, Springer, 2012. Besides providing full proof details, this revised and extended version improves running times, shows that Job Interval Selection is fixed-parameter tractable with respect to the standard parameter \(k\), and introduces the parameter \(c\)-compactness. Moreover, it adds an experimental evaluation of the algorithms.

Mathias Weller: Main work done while affiliated with Institut für Softwaretechnik und Theoretische Informatik, TU Berlin.

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van Bevern, R., Mnich, M., Niedermeier, R. et al. Interval scheduling and colorful independent sets. J Sched 18, 449–469 (2015). https://doi.org/10.1007/s10951-014-0398-5

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