A Faster Branching Algorithm for the Maximum $k$-Defective Clique Problem

TL;DR

This paper introduces a faster branching algorithm for the maximum $k$-defective clique problem, leveraging structural properties and conflict-based upper bounds, significantly improving performance over existing methods.

Contribution

The paper presents a novel branching algorithm that exploits structural properties and conflict relationships, achieving better asymptotic running time for the maximum $k$-defective clique problem.

Findings

Outperforms state-of-the-art solvers on benchmark datasets.

Introduces a new upper bound using conflict relationships.

Demonstrates improved efficiency through experimental results.

Abstract

A $k$-defective clique of an undirected graph $G$ is a subset of its vertices that induces a nearly complete graph with a maximum of $k$ missing edges. The maximum $k$-defective clique problem, which asks for the largest $k$-defective clique from the given graph, is important in many applications, such as social and biological network analysis. In the paper, we propose a new branching algorithm that takes advantage of the structural properties of the $k$-defective clique and uses the efficient maximum clique algorithm as a subroutine. As a result, the algorithm has a better asymptotic running time than the existing ones. We also investigate upper-bounding techniques and propose a new upper bound utilizing the \textit{conflict relationship} between vertex pairs. Because conflict relationship is common in many graph problems, we believe that this technique can be potentially generalized. Finally, experiments show that our algorithm outperforms state-of-the-art solvers on a wide range of open benchmarks.