A k-plex in a graph is a vertex set where each vertex is non-adjacent to at most k vertices (including itself) in this set, and the Maximum k-plex Problem (MKP) is to find the largest k-plex in the graph. 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. 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. In this paper, we first propose a new coloring-based upper bound, termed Relaxed Graph Color Bound (RelaxGCB), that significantly improves the previous coloring-based upper bound. 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 algorithms. Extensive experiments using diverse k values on hundreds of instances based on dense and massive sparse graphs demonstrate the excellent performance and robustness of our proposed methods.

翻译：图中的k-plex是指一个顶点集，其中每个顶点与该集合中至多k个顶点（包括自身）不相邻，而最大k-plex问题（MKP）旨在寻找图中最大的k-plex。作为一个NP困难的实际问题，MKP在诸多重要现实应用中发挥着作用，例如各类复杂网络的分析。分支定界（BnB）算法是MKP一类研究充分且有效的精确算法。近期针对MKP的BnB算法涉及两种分别基于图着色和划分的上界，它们从不同角度发挥作用，因此具有互补性。本文首先提出一种新的基于图着色的上界，称为松弛图着色界（RelaxGCB），该界显著改进了先前的基于图着色上界。我们进一步提出另一种新上界，称为松弛PUB（RelaxPUB），该界以新颖方式融合RelaxGCB与基于划分的上界，充分利用了二者的互补性。我们将RelaxGCB和RelaxPUB应用于目前最先进的BnB MKP算法，产生了八种新算法。基于密集和超大规模稀疏图的数百个实例上采用不同k值进行的广泛实验表明，我们提出的方法具有优异的性能和鲁棒性。