Statistical Mechanics of the Minimum Dominating Set Problem

TL;DR

This paper studies the minimum dominating set problem in networks, providing exact solutions for certain cases and near-optimal algorithms for more complex networks, with applications in network science.

Contribution

It introduces a generalized leaf-removal process and a mean-field theory with belief-propagation algorithms to solve the problem efficiently.

Findings

Exact solutions for networks without a core

Near-optimal solutions for networks with an extensive core

Algorithms perform well on real-world networks

Abstract

The minimum dominating set problem has wide applications in network science and related fields. It consists of assembling a node set of global minimum 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 is a difficult optimization problem in general, we show it can be exactly solved by a generalized leaf-removal process if the network contains no core. If the network has an extensive core, we estimate the size of minimum dominating sets by a mean-field theory and implement a belief-propagation algorithm to obtain near-optimal solutions. Our algorithms also perform well on real-world network instances.