Given an undirected graph $G$, a quasi-clique is a subgraph of $G$ whose density is at least $\gamma$ $(0 < \gamma \leq 1)$. Two optimization problems can be defined for quasi-cliques: the Maximum Quasi-Clique (MQC) Problem, which finds a quasi-clique with maximum vertex cardinality, and the Densest $k$-Subgraph (DKS) Problem, which finds the densest subgraph given a fixed cardinality constraint. Most existing approaches to solve both problems often disregard the requirement of connectedness, which may lead to solutions containing isolated components that are meaningless for many real-life applications. To address this issue, we propose two flow-based connectedness constraints to be integrated into known Mixed-Integer Linear Programming (MILP) formulations for either MQC or DKS problems. We compare the performance of MILP formulations enhanced with our connectedness constraints in terms of both running time and number of solved instances against existing approaches that ensure quasi-clique connectedness. Experimental results demonstrate that our constraints are quite competitive, making them valuable for practical applications requiring connectedness.

翻译：给定无向图 $G$，拟团是 $G$ 中密度至少为 $\gamma$ $(0 < \gamma \leq 1)$ 的子图。针对拟团可定义两个优化问题：最大拟团问题，即寻找顶点基数最大的拟团，以及最密 $k$-子图问题，即在固定基数约束下寻找密度最大的子团。现有解决这两个问题的大多数方法常忽略连通性要求，这可能导致解中包含孤立分量，而这些分量在许多实际应用中毫无意义。为解决该问题，我们提出两种基于流的连通性约束，可集成至针对 MQC 或 DKS 问题的已知混合整数线性规划（MILP）模型中。我们比较了增强连通性约束后的 MILP 公式在运行时间和求解实例数量方面与现有确保拟团连通性方法的性能。实验结果表明，我们的约束具有较强竞争力，使其在需要连通性的实际应用中具有重要价值。