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The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number

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Abstract

In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest. A well-developed example is the investigation of how the parameter treewidth influences the complexity of (other) graph problems. The reason why such investigations are of general interest is that real-world input distributions for computational problems often inherit structure from the natural computational processes that produce the problem instances (not necessarily in obvious, or well-understood ways). The max leaf number ml(G) of a connected graph G is the maximum number of leaves in a spanning tree for G. Exploring questions analogous to the well-studied case of treewidth, we can ask: how hard is it to solve 3-Coloring, Hamilton Path, Minimum Dominating Set, Minimum Bandwidth or many other problems, for graphs of bounded max leaf number? What optimization problems are W[1]-hard under this parameterization? We do two things:

  1. (1)

    We describe much improved FPT algorithms for a large number of graph problems, for input graphs G for which ml(G)≤k, based on the polynomial-time extremal structure theory canonically associated to this parameter. We consider improved algorithms both from the point of view of kernelization bounds, and in terms of improved fixed-parameter tractable (FPT) runtimes O *(f(k)).

  2. (2)

    The way that we obtain these concrete algorithmic results is general and systematic. We describe the approach, and raise programmatic questions.

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Authors and Affiliations

  1. University of Newcastle, Callaghan, Australia

    Michael Fellows & Frances Rosamond

  2. University of Bergen, Bergen, Norway

    Daniel Lokshtanov & Saket Saurabh

  3. The Institute of Mathematical Sciences, Chennai, 600113, India

    Neeldhara Misra

  4. Technical University of Eindhoven, Eindhoven, The Netherlands

    Matthias Mnich

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  1. Michael Fellows
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  2. Daniel Lokshtanov
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  3. Neeldhara Misra
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  4. Matthias Mnich
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  5. Frances Rosamond
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  6. Saket Saurabh
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Correspondence to Michael Fellows.

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Fellows, M., Lokshtanov, D., Misra, N. et al. The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number. Theory Comput Syst 45, 822–848 (2009). https://doi.org/10.1007/s00224-009-9167-9

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