Fast branching algorithm for Cluster Vertex Deletion

In the family of clustering problems, we are given a set of objects (vertices of the graph), together with some observed pairwise similarities (edges). The goal is to identify clusters of similar objects by slightly modifying the graph to obtain a cluster graph (disjoint union of cliques). Hueffner et al. [Theory Comput. Syst. 2010] initiated the parameterized study of Cluster Vertex Deletion, where the allowed modification is vertex deletion, and presented an elegant O(2^k * k^9 + n * m)-time fixed-parameter algorithm, parameterized by the solution size. In our work, we pick up this line of research and present an O(1.9102^k * (n + m))-time branching algorithm

Unifying Sparsest Cut, Cluster Deletion, and Modularity Clustering Objectives with Correlation Clustering

Graph clustering, or community detection, is the task of identifying groups of closely related objects in a large network. In this paper we introduce a new community-detection framework called LambdaCC that is based on a specially weighted version of correlation clustering. A key component in our methodology is a clustering resolution parameter, $\lambda$, which implicitly controls the size and structure of clusters formed by our framework. We show that, by increasing this parameter, our objective effectively interpolates between two different strategies in graph clustering: finding a sparse cut and forming dense subgraphs. Our methodology unifies and generalizes a number of other important clustering quality functions including modularity, sparsest cut, and cluster deletion, and places them all within the context of an optimization problem that has been well studied from the perspective of approximation algorithms. Our approach is particularly relevant in the regime of finding dense clusters, as it leads to a 2-approximation for the cluster deletion problem. We use our approach to cluster several graphs, including large collaboration networks and social networks

Even faster parameterized cluster deletion and cluster editing

Cluster Deletion and Cluster Editing ask to transform a graph by at most k edge deletions or edge edits, respectively, into a cluster graph, i.e., disjoint union of cliques. Equivalently, a cluster graph has no conflict triples, i.e., two incident edges without a transitive edge. We solve the two problems in time O(1.415^k) and O(1.76^k), respectively. These results round off our earlier work by considerably improved time bounds. For Cluster Deletion we use a technique that cuts away small connected components that do no longer contribute to the exponential part of the time complexity. As this idea is simple and versatile, it may lead to improvements for several other parameterized graph problems. The improvement for Cluster Editing is achieved by using the full power of an earlier structure theorem for graphs where no edge is in three conflict triples