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-Weightc-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.
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Ásgeirsson et al. (2017) interested in maximum-weight unions of c independent sets. In the graph theory literature, the problem is known as Max-Weight\(c\)-Colorable Subgraph; we prefer to stick to the established graph theory notion.
We are not aware that \(A\bowtie B\) or an alike notation has been used before. Rather, previous work has been coming up with ad hoc names for the classes \(A\bowtie B\) for various \(A\) and \(B\), often referring to one and the same class by several names.
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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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René van Bevern was supported by Grant 16-31-60007 mol_a_dk of the Russian Foundation for Basic Research. This work was initiated during a research visit of René van Bevern to TU Berlin in June 2017, partly supported by TU Berlin and by the Ministry of Science and Education of the Russian Federation under the 5-100 Excellence Programme.
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Bentert, M., van Bevern, R. & Niedermeier, R. Inductive \(k\)-independent graphs and c-colorable subgraphs in scheduling: a review. J Sched 22, 3–20 (2019). https://doi.org/10.1007/s10951-018-0595-8
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DOI: https://doi.org/10.1007/s10951-018-0595-8