The $K$-core of a graph is the unique maximum subgraph within which each vertex connects to $K$ or more other vertices. The optimal $K$-core attack problem asks to delete the minimum number of vertices from the $K$-core to induce its complete collapse. A hierarchical cycle-tree packing model is introduced here for this challenging combinatorial optimization problem. We convert the temporally long-range correlated $K$-core pruning dynamics into locally tree-like static patterns and analyze this model through the replica-symmetric cavity method of statistical physics. A set of coarse-grained belief propagation equations are derived to predict single vertex marginal probabilities efficiently. The associated hierarchical cycle-tree guided attack ({\tt hCTGA}) algorithm is able to construct nearly optimal attack solutions for regular random graphs and Erd\"os-R\'enyi random graphs. Our cycle-tree packing model may also be helpful for constructing optimal initial conditions for other irreversible dynamical processes on sparse random graphs.

翻译：图的$K$-核是满足每个顶点至少与$K$个其他顶点相连的唯一最大子图。最优$K$-核攻击问题要求删除$K$-核中尽可能少的顶点以使其完全瓦解。本文针对这一具有挑战性的组合优化问题，提出了一种层次化环树堆积模型。我们将时间上长程关联的$K$-核修剪动力学转化为局部树状静态模式，并通过统计物理的复制对称空腔方法分析该模型。推导出一组粗粒化置信传播方程，用于高效预测单顶点边际概率。基于该模型设计的层次化环树引导攻击算法({\tt hCTGA})能在规则随机图和Erdős-Rényi随机图上构建近乎最优的攻击方案。该环树堆积模型还可为稀疏随机图上其他不可逆动力学过程构建最优初始条件提供参考。