An Improved Fixed-Parameter Algorithm for 2-Club Cluster Edge Deletion

TL;DR

This paper presents an improved fixed-parameter algorithm for the 2-Club Cluster Edge Deletion problem, reducing the running time from previous algorithms to $O^*(2.695^k)$, enhancing efficiency in transforming graphs into disjoint unions of 2-clubs.

Contribution

The paper introduces a new fixed-parameter algorithm with a better running time for 2-Club Cluster Edge Deletion, fixing flaws in previous approaches.

Findings

Achieved a running time of $O^*(2.695^k)$ for the problem.

Corrected flaws in the previous $O^*(2.74^k)$ algorithm.

Provides a more efficient algorithm for graph modification to 2-clubs.

Abstract

A 2-club is a graph of diameter at most two. In the decision version of the parametrized {\sc 2-Club Cluster Edge Deletion} problem, an undirected graph $G$ is given along with an integer $k\geq 0$ as parameter, and the question is whether $G$ can be transformed into a disjoint union of 2-clubs by deleting at most $k$ edges. A simple fixed-parameter algorithm solves the problem in $\mathcal{O}^*(3^k)$, and a decade-old algorithm was claimed to have an improved running time of $\mathcal{O}^*(2.74^k)$ via a sophisticated case analysis. Unfortunately, this latter algorithm suffers from a flawed branching scenario. In this paper, an improved fixed-parameter algorithm is presented with a running time in $\mathcal{O}^*(2.695^k)$.