[Paper Review] Fast branching algorithm for Cluster Vertex Deletion

Anudhyan Boral, Marek Cygan+2 more|Warwick Research Archive Portal (University of Warwick)|Jun 17, 2013

Data Management and Algorithms22 references4 citations

TL;DR

This paper presents a new branching algorithm for the Cluster Vertex Deletion problem, achieving an improved time complexity of $\mathcal{O}(1.9102^k(n+m))$ by exploiting vertex cover structures in auxiliary graphs derived from candidate vertices. The method uses a refined case analysis and branching on vertex covers in $H_v$ graphs to reduce the branching factor below 2, significantly outperforming prior $\mathcal{O}(2^k k^9 + nm)$ algorithms.

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). 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.

Motivation & Objective

• To develop a faster fixed-parameter algorithm for Cluster Vertex Deletion, a problem central to graph clustering and correlation clustering.
• To improve upon the previous $\mathcal{O}(2^k k^9 + nm)$ algorithm by Hüffner et al. through a more efficient branching strategy.
• To achieve a sub-2 exponential dependency on the solution size $k$ by leveraging structural properties of the graph.
• To provide a linear-time dependency on graph size $n+m$ while minimizing the exponential term in $k$.
• To explore the potential for further improvements in FPT algorithms for Cluster Vertex Deletion, including kernelization.

Proposed method

• The algorithm uses a branching approach that identifies a vertex $v$ for which a solution exists not containing $v$, enabling reduction to a vertex cover problem on an auxiliary graph $H_v$.
• The auxiliary graph $H_v$ is constructed on vertices within distance 1 and 2 from $v$, with edges representing induced $P_3$ subgraphs in the original graph.
• The algorithm applies a specialized branching procedure on $H_v$ to find minimum vertex covers, using branching vectors such as $(1,2)$, $(1,3)$, and $(2,2)$.
• A key innovation is the use of greedy inclusion of vertices from $N_2$ (distance-2 neighbors) into the vertex cover to improve branching efficiency.
• The analysis distinguishes between marked and unmarked nodes in the branching tree to bound the worst-case branching vector, especially when $\texttt{MinVC}(H_v) \geq h$ for $h=2,3$.
• A Python script is used to enumerate and evaluate all possible branching vectors under different assumptions, enabling precise analysis of the worst-case branching factor.

Experimental results

Research questions

• RQ1 Can the exponential dependency on $k$ in Cluster Vertex Deletion be reduced below $2^k$ using a branching algorithm?
• RQ2 How can structural properties of the graph, particularly induced $P_3$ subgraphs, be exploited to design more efficient branching rules?
• RQ3 What is the optimal branching vector achievable when reducing Cluster Vertex Deletion to vertex cover on auxiliary graphs $H_v$?
• RQ4 Can the cost of identifying a suitable vertex $v$ to initiate the branching be balanced by the gains from efficient vertex cover branching on $H_v$?
• RQ5 What is the tightest possible upper bound on the branching factor for a fixed-parameter algorithm solving Cluster Vertex Deletion?

Key findings

• The paper achieves a time complexity of $\mathcal{O}(1.9102^k(n+m))$, representing a significant improvement over the previous $\mathcal{O}(2^k k^9 + nm)$ algorithm.
• The worst-case branching vector is bounded by $1.9102$, derived from the vector $(1,3,3,4,4,5)$, which corresponds to a specific sequence of branching steps on $H_v$.
• The analysis confirms that the branching vector $(1,3,3,4,4,5)$ yields the largest root of the characteristic equation less than $1.9102$, establishing the upper bound.
• The algorithm uses a novel strategy of marking nodes in the branching tree to analyze phases with high minimum vertex cover size, enabling tighter bounds.
• The method achieves linear dependency on $n+m$ while minimizing the exponential term, making it efficient for sparse graphs.
• The approach demonstrates that vertex cover-style branching on $H_v$ can be highly effective when combined with careful selection of the initial vertex $v$.