PhD Thesis

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 $(x,k) \in L$ in time $f(k) \cdot p(|x|)$, 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.