Hierarchical cycle-tree packing model for $K$-core attack problem

The $K$-core of a graph is the unique maximum subgraph within which each vertex connects to at least $K$ other vertices. The $K$-core optimal attack problem asks to construct a minimum-sized set of vertices whose removal results in the complete collapse of the $K$-core. In this paper, we construct a hierarchical cycle-tree packing model which converts a long-range correlated $K$-core pruning process into static patterns and analyze this model through the replica-symmetric (RS) cavity method of statistical physics. The cycle-tree guided attack (CTGA) message-passing algorithm exhibits superior performance on random regular and Erdos-Renyi graphs. It provides new upper bounds on the minimal cardinality of the $K$-core attack set. The model of this work may be extended to construct optimal initial conditions for other irreversible dynamical processes.