Exact Algorithms with New Upper Bounds for the Maximum k-plex Problem

The Maximum k-plex Problem (MKP) is a degree relaxation of the widely known Maximum Clique Problem. As a practical NP-hard problem, MKP has many important real-world applications, such as the analysis of various complex networks. Branch-and-bound (BnB) algorithms are a type of well-studied and effective exact algorithms for MKP, and the key for BnB algorithms is the bound design. Recent BnB MKP algorithms involve two kinds of upper bounds based on graph coloring and partition, respectively, that work in different perspectives and thus are complementary with each other. We first propose a new coloring-based upper bound, termed Relaxed Graph Color Bound (RelaxGCB), that significantly outperforms the previous coloring-based upper bound. Then we further propose another new upper bound, termed RelaxPUB, that incorporates RelaxGCB and a partition-based upper bound in a novel way, making use of their complementarity. We apply RelaxGCB and RelaxPUB to state-of-the-art BnB MKP algorithms and produce eight new BnB algorithms. Extensive experiments using diverse k values on hundreds of instances based on dense or massive sparse graphs demonstrate the excellent performance and robustness of our proposed methods.