Minimum Connected Dominating Set and Backbone of a Random Graph

TL;DR

This paper models the minimum connected dominating set problem in random graphs using a spin glass approach, deriving belief-propagation equations and developing an efficient message-passing algorithm for near-optimal solutions.

Contribution

It introduces a novel spin glass model with local constraints for the global connectivity problem and provides theoretical predictions and practical algorithms for random graphs.

Findings

Theoretical predictions match simulation results for regular and Erdős-Rényi graphs.

Developed an efficient message-passing algorithm for constructing near-minimum connected dominating sets.

The approach can be extended to other global topological constraints.

Abstract

We study the minimum dominating set problem as a representative combinatorial optimization challenge with a global topological constraint. The requirement that the backbone induced by the vertices of a dominating set should be a connected subgraph makes the problem rather nontrivial to investigate by statistical physics methods. Here we convert this global connectivity constraint into a set of local vertex constraints and build a spin glass model with only five coarse-grained vertex states. We derive a set of coarse-grained belief-propagation equations and obtain theoretical predictions on the relative sizes of minimum dominating sets for regular random and Erd\"os-R\'enyi random graph ensembles. We also implement an efficient message-passing algorithm to construct close-to-minimum connected dominating sets and backbone subgraphs for single random graph instances. Our theoretical strategy may also be inspiring for some other global topological constraints.