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Inductive \(k\)-independent graphs and c-colorable subgraphs in scheduling: a review

  • Matthias Bentert
  • René van BevernEmail author
  • Rolf Niedermeier
Article

Abstract

Inductive \(k\)-independent graphs generalize chordal graphs and have recently been advocated in the context of interference-avoiding wireless communication scheduling. The NP-hard problem of finding maximum-weight induced c-colorable subgraphs, which is a generalization of finding maximum independent sets, naturally occurs when selecting \(c\) sets of pairwise non-conflicting jobs (modeled as graph vertices). We investigate the parameterized complexity of this problem on inductive \(k\)-independent graphs. We show that the Maximum Independent Set problem is W[1]-hard even on 2-simplicial 3-minoes—a subclass of inductive 2-independent graphs. In contrast, we prove that the more general Max-Weight c-Colorable Subgraph problem is fixed-parameter tractable on edge-wise unions of cluster and chordal graphs, which are 2-simplicial. In both cases, the parameter is the solution size. Aside from this, we survey other graph classes between inductive \(1\)-independent and inductive \(2\)-independent graphs with applications in scheduling.

Keywords

Independent set Job interval selection Interval graphs Chordal graphs Inductive \(k\)-independent graphs NP-hard problems Parameterized complexity 

Notes

Acknowledgements

We are grateful to Andreas Krebs (Tübingen) for fruitful discussions concerning parts of this work. We thank the anonymous referees of Journal of Scheduling for constructive feedback.

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Authors and Affiliations

  1. 1.Algorithmics and Computational Complexity, Faculty IVTU BerlinGermany
  2. 2.Department of Mechanics and MathematicsNovosibirsk State UniversityNovosibirskRussian Federation
  3. 3.Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of SciencesNovosibirskRussian Federation

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