Statistical Mechanics of the Minimum Dominating Set Problem

The minimum dominating set (MDS) problem has wide applications in network science and related fields. It aims at constructing a node set of smallest size such that any node of the network is either in this set or is adjacent to at least one node of this set. Although this optimization problem is generally very difficult, we show it can be exactly solved by a generalized leaf-removal (GLR) process if the network contains no core. We present a percolation theory to describe the GLR process on random networks, and solve a spin glass model by mean field method to estimate the MDS size. We also implement a message-passing algorithm and a local heuristic algorithm that combines GLR with greedy node-removal to obtain near-optimal solutions for single random networks. Our algorithms also perform well on real-world network instances.