Dense subgraph extraction is a fundamental problem in graph analysis and data mining, aimed at identifying cohesive and densely connected substructures within a given graph. It plays a crucial role in various domains, including social network analysis, biological network analysis, recommendation systems, and community detection. However, extracting a subgraph with the highest node similarity is a lack of exploration. To address this problem, we studied the Member Selection Problem and extended it with a dynamic constraint variant. By incorporating dynamic constraints, our algorithm can adapt to changing conditions or requirements, allowing for more flexible and personalized subgraph extraction. This approach enables the algorithm to provide tailored solutions that meet specific needs, even in scenarios where constraints may vary over time. We also provide the theoretical analysis to show that our algorithm is 1/3-approximation. Eventually, the experiments show that our algorithm is effective and efficient in tackling the member selection problem with dynamic constraints.

翻译：密集子图提取是图分析和数据挖掘中的一个基础问题，旨在识别给定图中具有内聚性和高连通性的子结构。它在社交网络分析、生物网络分析、推荐系统和社区检测等多个领域发挥着关键作用。然而，提取具有最高节点相似度的子图尚未得到充分探索。为解决该问题，我们研究了成员选择问题，并扩展了其带动态约束的变体。通过引入动态约束，我们的算法能够适应不断变化的条件或需求，从而实现更灵活、个性化的子图提取。该方法使算法能够提供满足特定需求的定制化解决方案，即使在约束条件随时间变化的场景下也能适用。我们还提供了理论分析，证明该算法具有1/3近似比。最终，实验表明，我们的算法在处理带动态约束的成员选择问题上既有效又高效。