In the Shortest Superstring problem we are given a set of strings S = {s1,..., sn} and an integer ` and the question is to decide whether there is a superstring s of length at most ` containing all strings of S as substrings. We obtain several parameterized algorithms and complexity results for this problem. In particular, we give an algorithm which in time 2O(k) poly(n) finds a superstring of length at most ` containing at least k strings of S. We complement this by the lower bound showing that such a parameterization does not admit a polynomial kernel up to some complexity assumption. We also obtain several results about “below guaranteed values” parameterization of the problem. We show that parameterization by compression admits a polynomial kernel while parameterization “below matching ” is hard.

In graph modification problems, one is given a graph G and the goal is to apply a minimum number of modification operations (such as edge deletions) to G such that the resulting graph fulfills a certain property. For example, the Cluster Deletion problem asks to delete as few edges as possible such that the resulting graph is a disjoint union of cliques. Graph modification problems appear in numerous applications, including the analysis of biological and social networks. Typically, graph modification problems are NP-hard, making them natural candidates for parameterized complexity studies. We discuss several fruitful interactions between the development of fixed-parameter algorithms and the design of heuristics for graph modification problems, featuring quite different aspects of mutual benefits.