# Minimal Dominating Set problem studied by simulated annealing and cavity method: Analytics and population dynamics

## TL;DR

This paper investigates the minimal dominating set problem using advanced statistical physics methods and algorithms, revealing complex solution space transitions and developing effective algorithms for estimating ground state energies.

## Contribution

It extends the cavity method with replica symmetry breaking to analyze the solution space and introduces survey propagation algorithms for better ground state estimation.

## Key findings

- Solution space exhibits condensation and cluster transitions.
- Survey propagation algorithms effectively estimate ground state energy.
- Developed algorithms outperform some existing methods.

## Abstract

The minimal dominating Set (MDS) problem is a prototypical hard combinatorial optimization problem. Two years ago we studied this problem by cavity method. Although we get the solution of a given graph, which gives very good estimation of minimal dominating size, but we don't know whether we get the ground state solution and how many solutions exist in the ground state. For this purpose, last year we continue to develop the one step replica symmetry breaking (RSB) theory to find the ground state energy of the MDS problem. Finally we find that 1) The MDS problem solution space has both condensation transition and cluster transition on regular Random (RR) graph and we prove this by simulated annealing dynamical process. 2) We developed zero temperature Survey Propagation (SP) algorithm on ER graph to estimate the ground state energy and to get Survey Propagation Decimation (SPD) algorithm with good results same as BPD algorithm.