The Directed Dominating Set Problem: Generalized Leaf Removal and Belief Propagation

TL;DR

This paper introduces algorithms and theories for approximating minimum dominating sets in directed graphs, combining generalized leaf removal and belief propagation, applicable to both random and real-world networks.

Contribution

It presents a novel combination of local and message-passing algorithms with theoretical analysis for the directed dominating set problem.

Findings

Algorithms achieve near-optimal solutions for random digraphs.

Theoretical models explain the structure of solutions.

Methods are effective on real-world network instances.

Abstract

A minimum dominating set for a digraph (directed graph) is a smallest set of vertices such that each vertex either belongs to this set or has at least one parent vertex in this set. We solve this hard combinatorial optimization problem approximately by a local algorithm of generalized leaf removal and by a message-passing algorithm of belief propagation. These algorithms can construct near-optimal dominating sets or even exact minimum dominating sets for random digraphs and also for real-world digraph instances. We further develop a core percolation theory and a replica-symmetric spin glass theory for this problem. Our algorithmic and theoretical results may facilitate applications of dominating sets to various network problems involving directed interactions.