KD-Club: An Efficient Exact Algorithm with New Coloring-based Upper Bound for the Maximum k-Defective Clique Problem

TL;DR

This paper introduces KD-Club, an exact algorithm for the Maximum k-Defective Clique Problem that employs a novel coloring-based upper bound to improve search efficiency and solve more instances faster.

Contribution

It proposes a new coloring-based upper bound (CLUB) and integrates it into a branch-and-bound algorithm, significantly enhancing performance over existing methods.

Findings

KD-Club solves more instances within time limits.

It has a smaller search tree and shorter solving times.

Outperforms state-of-the-art algorithms on benchmarks.

Abstract

The Maximum k-Defective Clique Problem (MDCP) aims to find a maximum k-defective clique in a given graph, where a k-defective clique is a relaxation clique missing at most k edges. MDCP is NP-hard and finds many real-world applications in analyzing dense but not necessarily complete subgraphs. Exact algorithms for MDCP mainly follow the Branch-and-bound (BnB) framework, whose performance heavily depends on the quality of the upper bound on the cardinality of a maximum k-defective clique. The state-of-the-art BnB MDCP algorithms calculate the upper bound quickly but conservatively as they ignore many possible missing edges. In this paper, we propose a novel CoLoring-based Upper Bound (CLUB) that uses graph coloring techniques to detect independent sets so as to detect missing edges ignored by the previous methods. We then develop a new BnB algorithm for MDCP, called KD-Club, using CLUB in both the preprocessing stage for graph reduction and the BnB searching process for branch pruning. Extensive experiments show that KD-Club significantly outperforms state-of-the-art BnB MDCP algorithms on the number of solved instances within the cut-off time, having much smaller search tree and shorter solving time on various benchmarks.