The Directed Dominating Set problem studied by cavity method: Warning propagation and population dynamics

TL;DR

This paper extends the cavity method and replica symmetry breaking theory to analyze the directed minimal dominating set problem, revealing phase transition thresholds and developing algorithms with promising results for Erdős-Rényi graphs.

Contribution

It advances the theoretical understanding of the directed minimal dominating set problem using cavity method and develops effective algorithms like survey propagation decimation.

Findings

Warning propagation fails to converge above a connectivity of 3.704.

Ground state energy and transition points are identified for Erdős-Rényi graphs.

Survey propagation decimation performs comparably to belief propagation decimation.

Abstract

The minimal dominating set for a digraph(directed graph)is a prototypical hard combinatorial optimization problem. In a previous paper, we studied this problem using the cavity method. Although we found a solution for a given graph that gives very good estimate of the minimal dominating size, we further developed the one step replica symmetry breaking theory to determine the ground state energy of the undirected minimal dominating set problem. The solution space for the undirected minimal dominating set problem exhibits both condensation transition and cluster transition on regular random graphs. We also developed the zero temperature survey propagation algorithm on undirected Erd\H{o}s-R\'enyi graphs to find the ground state energy. In this paper we continue to develop the one step replica symmetry breaking theory to find the ground state energy for the directed minimal dominating set problem. We find the following. (1)The warning propagation equation can not converge when the connectivity is greater than the core percolation threshold value of 3.704. Positive edges have two types warning, but the negative edges have one. (2)We determine the ground state energy and the transition point of the Erd\H{o}s-R\'enyi random graph. (3)The survey propagation decimation algorithm has good results comparable with the belief propagation decimation algorithm. Keywords: directed minimal dominating set , replica symmetry breaking, Erd\H{o}s-R\'enyi graph, warning propagation, survey propagation decimation.