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|Title:||Kernels for the F deletion problem|
|University:||Homi Bhabha National Institute|
|Completed Date:||September 2011|
|Abstract:||In this thesis, we use the parameterized framework for the design and analysis of algorithms for NP-complete problems. This amounts to studying the parameterized version of the classical decision version. Herein, the classical language appended with a secondary measure called a ?parameter?. The central notion in parameterized complexity is that of fixed-parameter tractability, which means given an instance (x, k) of a parameterized language L, deciding whether (x, k) ! L in time f(k) ?p(|x|), where f is an arbitrary function of k alone and p is a polynomial function. The notion of kernelization formalizes preprocessing or data reduction, and refers to polynomial time algorithms that transform any given input into an equivalent instance whose size is bounded as a function of the parameter alone. The center of our attention in this thesis is the F-Deletion problem, a vastly general question that encompasses many fundamental optimization problems as special cases. In particular, we provide evidence supporting a conjecture about the kernelization complexity of the problem, and this work branches off in a number of directions, leading to results of independent interest. We also study the Colorful Motifs problem, a well-known question that arises frequently in practice. Our investigation demonstrates the hardness of the problem even when restricted to very simple graph classes. The F-Deletion Problem Let F be a finite family of graphs. The F-Deletion problem takes as input a graph G on n vertices, and a positive integer k. The question is whether it is possible to delete at most k vertices from G such that the remaining graph contains no graph from F as a minor. This question encompasses fundamental problems such as Vertex Cover (consider F consisting of the graph with two vertices and one edge) or Feedback Vertex Set (the set F consists of a cycle with three vertices). A number of other deletion-based optimization problems also turn out to be special cases of Planar F-Deletion.|
|Appears in Departments:||Department of Mathematical Sciences|
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