$k$-plexes relax cliques by allowing each vertex to disconnect to at most $k$ vertices. Finding a maximum $k$-plex in a graph is a fundamental operator in graph mining and has been receiving significant attention from various domains. The state-of-the-art algorithms all adopt the branch-reduction-and-bound (BRB) framework where a key step, called reduction-and-bound (RB), is used for narrowing down the search space. A common practice of RB in existing works is SeqRB, which sequentially conducts the reduction process followed by the bounding process once at a branch. However, these algorithms suffer from the efficiency issues. In this paper, we propose a new alternated reduction-and-bound method AltRB for conducting RB. AltRB first partitions a branch into two parts and then alternatively and iteratively conducts the reduction process and the bounding process at each part of a branch. With newly-designed reduction rules and bounding methods, AltRB is superior to SeqRB in effectively narrowing down the search space in both theory and practice. Further, to boost the performance of BRB algorithms, we develop efficient and effective pre-processing methods which reduce the size of the input graph and heuristically compute a large $k$-plex as the lower bound. We conduct extensive experiments on 664 real and synthetic graphs. The experimental results show that our proposed algorithm kPEX with AltRB and novel pre-processing techniques runs up to two orders of magnitude faster and solves more instances than state-of-the-art algorithms.

翻译：$k$-plex通过允许每个顶点至多与$k$个顶点不连通来松弛团的概念。在图中寻找最大$k$-plex是图挖掘中的基本操作，并持续受到多个领域的广泛关注。现有最先进的算法均采用分支-约简-定界框架，其中关键步骤——称为约简-定界——用于缩小搜索空间。现有工作中约简-定界的常规实现是SeqRB，即在每个分支顺序执行一次约简过程后接定界过程。然而，这些算法存在效率问题。本文提出一种新的交替约简-定界方法AltRB来执行约简-定界操作。AltRB首先将分支划分为两部分，随后在每个分支部分交替迭代地执行约简过程与定界过程。通过新设计的约简规则与定界方法，AltRB在理论与实践中均优于SeqRB，能更有效地缩小搜索空间。此外，为提升分支-约简-定界算法的性能，我们开发了高效且有效的预处理方法，通过缩减输入图规模并启发式地计算大型$k$-plex作为下界。我们在664个真实与合成图上进行了大量实验。实验结果表明，采用AltRB及新型预处理技术的算法kPEX，其运行速度比现有最先进算法快达两个数量级，并能求解更多实例。