The Maximum s-Bundle Problem (MBP) addresses the task of identifying a maximum s-bundle in a given graph. A graph G=(V, E) is called an s-bundle if its vertex connectivity is at least |V|-s, where the vertex connectivity equals the minimum number of vertices whose deletion yields a disconnected or trivial graph. MBP is NP-hard and holds relevance in numerous realworld scenarios emphasizing the vertex connectivity. Exact algorithms for MBP 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 s-bundle and the initial lower bound with graph reduction. In this work, we introduce a novel Partition-based Upper Bound (PUB) that leverages the graph partitioning technique to achieve a tighter upper bound compared to existing ones. To increase the lower bound, we propose to do short random walks on a clique to generate larger initial solutions. Then, we propose a new BnB algorithm that uses the initial lower bound and PUB in preprocessing for graph reduction, and uses PUB in the BnB search process for branch pruning. Extensive experiments with diverse s values demonstrate the significant progress of our algorithm over state-of-the-art BnB MBP algorithms. Moreover, our initial lower bound can also be generalized to other relaxation clique problems.

翻译：最大$s$-束团问题(MBP)要求从给定图中寻找最大$s$-束团。图$G=(V,E)$被称为$s$-束团，当且仅当其顶点连通度至少为$|V|-s$，其中顶点连通度定义为删除后使图变为非连通图或平凡图所需的最小顶点数目。MBP是NP难问题，在众多强调顶点连通度的现实场景中具有重要应用价值。针对MBP的精确算法主要遵循分支定界(BnB)框架，其性能严重依赖于最大$s$-束团基数上界的质量以及结合图归约的初始下界。本文提出一种新颖的基于划分的上界(PUB)，通过利用图划分技术获得比现有方法更紧致的上界。为提升下界，我们提出在团上执行短随机游走以生成更大的初始解。进而提出新型BnB算法：在预处理阶段利用初始下界和PUB进行图归约，在BnB搜索过程中使用PUB实现分支剪枝。针对不同$s$值的广泛实验表明，本算法相较当前最先进的BnB MBP算法取得了显著进展。此外，所提出的初始下界方法可推广至其他松弛团问题。