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

  • Authors:

  • Michael Fellows;Frances Rosamond

  • Affiliations:

  • University of Newcastle, Callaghan NSW 2308, Australia and Durham University, Institute of Advanced Study, Durham DH1 3RL, United Kingdom;University of Newcastle, Callaghan NSW 2308, Australia

  • Venue:
  • CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
  • Year:
  • 2007

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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 treewidthinfluences 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 numberof a connected graph Gis 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-Coloringor Hamilton Pathor Minimum Dominating Setfor graphs of bounded max leaf number? We do two things:(1) We describe much improved FPT algorithms for a large number of graph problems, for input of bounded max leaf number, based on the polynomial-time extremal structure theory associated to the parameter max leaf number.(2) The way that we obtain these concrete algorithmic results is general and systematic. We describe the approach.