A Fast Branching Algorithm for Cluster Vertex Deletion

Abstract

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).Hüffner 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 + nm)-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.

On Happy Colorings, Cuts, and Structural Parameterizations

“…Let (G, p, k, S) be an instance of MHV, and S is a given minimum modulator to cluster of G. We describe an algorithm that works in O * (d d ), where d = |S| is the distance to cluster parameter of G. Note that it is not necessary that S is given explicitly. To find S, one can simply consider G as an instance of Cluster Vertex Deletion parameterized by the solution size, and employ one of the algorithms working in time O * (c d ), let it be a simple O * (3 d ) running time algorithm [16], or more sophisticated ones, working in O * (2 d ) [15] or even in O * (1.9102 d ) [5] running time. Note that this would not change the overall…”

Faster parameterized algorithm for pumpkin vertex deletion set

“…Lemma 1 (Boral et al [1]). Let F be a dominating family of v. There is a cluster deletion set S of G of minimum size such that either v ∈ S or there is X ∈ F such that X ⊆ S.…”

“…Later, Hüffner et al [8] gave an O * (2 k )time algorithm for Cluster Vertex Deletion based on iterative compression. Finally, Boral et al [1] gave an O * (1.911 k )-time algorithm.…”

“…Our algorithm is based on the algorithm for Boral et al [1]. The algorithm of Boral et al works as follows.…”

A Refined Complexity Analysis of Finding the Most Vital Edges for Undirected Shortest Paths

“…x · (n + m)) time [6]. Our algorithm is based on the observation that twins can be handled equally in a solution.…”