The Kernelization Complexity of Some Domination and Covering Problems

P olynomial-time preprocessing is a simple algorithmic strategy which has been widely employed in practice to tackle hard problems. e quantication and analysis of the e›ciency of preprocessing algorithms are, in a certain precise sense, outside the pale of classical complexity theory. e notion of kernelization from parameterized complexity theory provides a framework for the mathematical analysis of polynomial-time preprocessing algorithms. Both kernelization and the closely related notion of xed-parameter tractable (FPT) algorithms are very active areas of current research. In this thesis we describe the results of our study of the kernelization complexity of some graph domination and covering problems. An instance of a parameterized problem is of the form (x , k) where x is a classical problem instance and k is a suitably-chosen parameter. A xed-parameter tractable (FPT) algorithm for the problem is an algorithm which solves the problem in timeO( f (k) ⋅ ∣x∣c) for some computable function f () and constant c. A kernelization algorithm for the problem is a polynomial-time algorithm which converts the input instance (x , k) to an equivalent parameterized instance (x′, k′) where both the size of the new instance x′ and the value of the new parameter k′ are bounded by some computable function f (k) of the original parameter k. e new instance is called a kernel for the problem, and f (k) is the size of the kernel. In the following, n denotes the number of vertices in the input graph, and k is the parameter in each case. e study of variants of the domination problem in graphs has been a vibrant area of research for many decades, and continues to be a rich source of graph-theoretical and algorithmic problems. e prototypical problem in this eld is Dominating Set, a classical minimization problem which asks whether the input graph has a dominating