Graph-Modeled Data Clustering

ly speaking, what have we done in the previous section? After applying a number of rules in polynomial time to an instance of VERTEX COVER, we arrived at a reduced instance whose size can solely be expressed in terms of the parameter k. Since this can be easily done in O(n) time, we have found a data reduction for VERTEX COVER with guarantees concerning its running time as well as its effectiveness. These properties are formalized in the concepts of a problem kernel and the corresponding kernelization [25]. Definition 1.2. Let L be a parameterized problem, that is, L consists of input pairs (I, k), where I is the problem instance and k is the parameter. A reduction to a problem kernel (or kernelization) means to replace an instance (I, k) by a reduced instance (I ′, k′) called problem kernel in polynomial time such that (1) k′ ≤ k, (2) I ′ is smaller than g(k) for some function g only depending on k, and (3) (I, k) has a solution if and only if (I ′, k′) has one. While this definition does not formally require that it is possible to reconstruct a solution for the original instance from a solution for the problem kernel, all kernelizations we are aware of easily allow for this. The methodological approach of kernelization, including various techniques of data reduction, is best learned by the concrete examples that we discuss in Sec. 1.3; there, we will also discuss kernelizations for VERTEX COVER that even yield a kernel with a linear number of vertices in k. To conclude this section, we state some useful general observations and remarks concerning Definition 1.2 and its connections to fixed-parameter tractability. Most notably, there is a close connection between fixed-parameter tractable problems and those problems that have a problem kernel—they are exactly the same. Theorem 1.1 (Cai et al. [11]). Every fixed-parameter tractable problem is kernelizable and vice-versa. Unfortunately, the practical use of this theorem is limited: the running times of a fixed-parameter algorithm directly obtained from a kernelization is usually not practical; and, in the other direction, the theorem does not constructively provide us with a data reduction scheme for a fixed-parameter tractable problem. Hence, the main use of Theorem 1.1 is to establish the fixed-parameter tractability or amenability to kernelization of a problem—or show that we need not search any