The Edge Interdiction Clique Problem (EICP) aims to remove at most $k$ edges from a graph so as to minimize the size of the largest clique in the remaining graph. This problem captures a fundamental question in graph manipulation: which edges are structurally critical for preserving large cliques? Such a problem is also motivated by practical applications including protein function maintenance and image matching. The EICP is computationally challenging and belongs to a complexity class beyond NP. Existing approaches rely on general mixed-integer bilevel programming solvers or reformulate the problem into a single-level mixed integer linear program. However, they are still not scalable when the graph size and interdiction budget $k$ grow. To overcome this, we investigate new mixed integer linear formulations, which recast the problem into a sequence of parameterized Edge Blocker Clique Problems (EBCP). This perspective decomposes the original problem into simpler subproblems and enables tighter modeling of clique-related inequalities. Furthermore, we propose a two-stage exact algorithm, \textsc{RLCM}, which first applies problem-specific reduction techniques to shrink the graph and then solves the reduced problem using a tailored branch-and-cut framework. Extensive computational experiments on maximum clique benchmark graphs, large real-world sparse networks, and random graphs demonstrate that \textsc{RLCM} consistently outperforms existing approaches.

翻译：边阻断团问题（EICP）旨在从图中移除至多 $k$ 条边，以最小化剩余图中最大团的规模。该问题捕捉了图操控中的一个基本问题：哪些边在结构上对于维持大团至关重要？此类问题也受到蛋白质功能维持和图像匹配等实际应用的推动。EICP 在计算上具有挑战性，属于超越 NP 的复杂性类别。现有方法依赖于通用的混合整数双层规划求解器，或将问题重构为单层混合整数线性规划。然而，当图规模和阻断预算 $k$ 增长时，这些方法仍然难以扩展。为克服此问题，我们研究了新的混合整数线性规划模型，将问题重新表述为一系列参数化的边阻断团问题（EBCP）。这一视角将原问题分解为更简单的子问题，并实现了对团相关不等式的更紧致建模。此外，我们提出了一种两阶段精确算法 \textsc{RLCM}，该算法首先应用特定于问题的归约技术来收缩图，然后使用定制的分支切割框架求解归约后的问题。在最大团基准图、大型现实世界稀疏网络和随机图上的大量计算实验表明，\textsc{RLCM} 始终优于现有方法。