Cluster deletion revisited

In the weighted Cluster Deletion problem we are given a graph with non-negative integral edge weights and the task is to determine, for a target value k, if there is a set of edges of total weight at most k such that its removal results in a disjoint union of cliques. It is well-known that the problem is FPT parameterized by k, the total weight of edge deletions. In scenarios in which the solution size is large, naturally one needs to drop the constraint on the solution size. Here we study weighted Cluster Deletion where there is no bound on the size of the solution, but the parameter captures structural properties of the input graph. Our main contribution is to classify the parameterized complexity of weighted Cluster Deletion with three structural parameters, namely, vertex cover, twin cover and neighborhood diversity. We show that the problem is FPT when parameterized by the vertex cover, whereas it becomes paraNP-hard when parameterized by the twin cover or the neighborhood diversity. To illustrate the applicability of our FPT result, we use it in order to show that the unweighted variant of the problem, Cluster Deletion, is FPT parameterized by the twin cover. This is the first algorithm with single-exponential running time parameterized by the twin cover. Interestingly, we are able to achieve an FPT result parameterized by the neighborhood diversity that involves an ILP formulation. In fact, our results generalize the parameterized setting by the solution size, as we deduce that both parameters, twin cover and neighborhood diversity, are linearly bounded by the number of edge deletions.

Parameterized and exact algorithms for class domination coloring

Large graphs are prevalent in many applications and enable a variety of information dissemination processes, e.g., meme, virus, and influence propagation. How can we optimize the underlying graph structure to affect the outcome of such dissemination processes in a desired way (e.g., stop a virus propagation, facilitate the propagation of a piece of good idea, etc)? Existing research suggests that the leading eigenvalue of the underlying graph is the key metric in determining the so-called epidemic threshold for a variety of dissemination models. In this paper, we study the problem of how to optimally place a set of edges (e.g., edge deletion and edge addition) to optimize the leading eigenvalue of the underlying graph, so that we can guide the dissemination process in a desired way. We propose effective, scalable algorithms for edge deletion and edge addition, respectively. In addition, we reveal the intrinsic relationship between edge deletion and node deletion problems. Experimental results validate the effectiveness and efficiency of the proposed algorithms.

Split Vertex Deletion meets Vertex Cover: New fixed-parameter and exact exponential-time algorithms

A counterexample to a dynamic algorithm for convex hulls of line arrangements

On the Parameterized Complexity of s-club Cluster Deletion Problems

Many fixed-parameter tractable algorithms using a bounded search tree have been repeatedly improved, often by describing a larger number of branching rules involving an increasingly complex case analysis. We introduce a novel and general search strategy that branches on the forbidden subgraphs of a graph class relaxation. By using the class of P4-sparse graphs as the relaxed graph class, we obtain efficient bounded search tree algorithms for several parametrized deletion problems. We give the first non-trivial bounded search tree algorithms for the cograph edge-deletion problem and the trivially perfect edge-deletion problems. For the cograph vertex deletion problem, a refined analysis of the runtime of our simple bounded search algorithm gives a faster exponential factor than those algorithms designed with the help of complicated case distinctions and non-trivial running time analysis [R. Niedermeier and P. Rossmanith, An efficient fixed-parameter algorithm for 3-hitting set, J. Discrete Algorithms1(1) (2003) 89–102] and computer-aided branching rules [J. Gramm, J. Guo, F. Hüffner and R. Niedermeier, Automated generation of search tree algorithms for hard graph modification problems, Algorithmica39(4) (2004) 321–347].