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Inductive \(k\)-independent graphs and \(c\)-colorable subgraphs in scheduling: a review. (English) Zbl 1425.90039
Summary: 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.
90B35 Deterministic scheduling theory in operations research
68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
90C35 Programming involving graphs or networks
05C15 Coloring of graphs and hypergraphs
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