Frontiers in Algorithmics

We give an overview of recent results and techniques in parameterized algorithms for graph modification problems. In network (or graph) modifications problem we have to modify (repair, improve, or adjust) a network to satisfy specific required properties while keeping the cost of the modification to the minimum. The commonly adapted mathematical model in the study of network problems is the graph modification problem. This is a fundamental unifying problem with a tremendous number of applications in various disciplines like machine learning, networking, sociology, data mining, computational biology, computer vision, and numerical analysis, and many others. We start with three generic examples of graph modification. Our first example is the connectivity augmentation problem. Here the graph models an existing network (say, a telecommunication network) and the goal is to enhance the network to ensure resilience against link failures. In other words, by adding a few links between nodes we wish to obtain a network with better connectivity. This is a special case of the graph modification problem where we want to improve the connectivity of a graph. The second example is graph clustering. This is the fundamental problem of identification of closely related objects that have many interactions within themselves and few with the rest of the system. A group of objects of this type is known as a cluster (or community). In Fig. 1 one can find a clustering of the scientific-collaboration network of the members of the Algorithms groups at the University of Bergen. One of the most popular common to clustering is to model a system as a weighted graph. Then the task is to identify a set of low-cost edges (insignificant interactions) which removal partition the graph into clusters and this is again a special case of the graph modification problem. The third example is the fundamental problem arising in sparse matrix computations. During Gaussian eliminations of large sparse matrices new non-zero elements called fill can replace original zeros thus increasing storage requirements and running time needed to solve the system. The problem of finding the right Supported by the European Research Council (ERC) via grant Rigorous Theory of Preprocessing, reference 267959. c © Springer International Publishing Switzerland 2015 J. Wang and C. Yap (Eds.): FAW 2015, LNCS 9130, pp. 3–6, 2015. DOI: 10.1007/978-3-319-19647-3 1