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. MKP is a practical NP-hard problem, and the k-plex model 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. Then we propose another new upper bound, termed SeesawUB, inspired by the seesaw playing game, that incorporates RelaxGCB and a partition-based upper bound in a novel way, making use of their complementarity. We apply RelaxGCB and SeesawUB to state-of-the-art BnB MKP algorithms and produce four new algorithms. Extensive experiments show the excellent performance and robustness of the new algorithms with our proposed upper bounds.

翻译：图中的k-plex是指一个顶点集合，其中每个顶点与该集合内至多k个顶点（包括自身）不相邻，最大k-plex问题（MKP）旨在寻找图中最大的k-plex。MKP是一个实际的NP难问题，k-plex模型在诸多重要实际应用中具有关键作用，例如各类复杂网络的分析。分支定界（BnB）算法是一类经过充分研究且有效的MKP精确算法。近期基于BnB的MKP算法分别引入了基于图着色和基于划分的两类上界，它们从不同视角发挥作用，因而具有互补性。本文首先提出一种新型基于着色的上界——松弛图着色界（RelaxGCB），显著改进了原有的基于着色上界。随后，受跷跷板游戏启发，我们提出另一个新上界——跷跷板界（SeesawUB），它以新颖方式融合了RelaxGCB与基于划分的上界，充分利用了二者的互补性。我们将RelaxGCB与SeesawUB应用于当前最先进的BnB MKP算法，生成了四种新算法。大量实验表明，采用我们提出的上界的新算法具有优异的性能与鲁棒性。