Theory of Computing Systems

, Volume 45, Issue 4, pp 822–848 | Cite as

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

  • Michael Fellows
  • Daniel Lokshtanov
  • Neeldhara Misra
  • Matthias Mnich
  • Frances Rosamond
  • Saket Saurabh


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.



Parameterized complexity Max-leaf Bandwidth Well-quasiordering Kernelization 


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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Michael Fellows
    • 1
  • Daniel Lokshtanov
    • 2
  • Neeldhara Misra
    • 3
  • Matthias Mnich
    • 4
  • Frances Rosamond
    • 1
  • Saket Saurabh
    • 2
  1. 1.University of NewcastleCallaghanAustralia
  2. 2.University of BergenBergenNorway
  3. 3.The Institute of Mathematical SciencesChennaiIndia
  4. 4.Technical University of EindhovenEindhovenThe Netherlands

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