# An Order-based Algorithm for Minimum Dominating Set with Application in Graph Mining

## TL;DR

This paper introduces a new order-based randomized local search algorithm for efficiently finding minimum dominating sets in large graphs, outperforming existing methods and applicable to various real-world network types.

## Contribution

The paper presents a novel order-based RLS algorithm for the minimum dominating set problem, demonstrating superior performance on large and diverse graphs.

## Key findings

- RLS_o outperforms classical greedy and metaheuristic algorithms.
- Effective in large graphs with tens of thousands of vertices.
- Multi-start RLS_o suitable for minimum weight dominating set.

## Abstract

Dominating set is a set of vertices of a graph such that all other vertices have a neighbour in the dominating set. We propose a new order-based randomised local search (RLS$_o$) algorithm to solve minimum dominating set problem in large graphs. Experimental evaluation is presented for multiple types of problem instances. These instances include unit disk graphs, which represent a model of wireless networks, random scale-free networks, as well as samples from two social networks and real-world graphs studied in network science. Our experiments indicate that RLS$_o$ performs better than both a classical greedy approximation algorithm and two metaheuristic algorithms based on ant colony optimisation and local search. The order-based algorithm is able to find small dominating sets for graphs with tens of thousands of vertices. In addition, we propose a multi-start variant of RLS$_o$ that is suitable for solving the minimum weight dominating set problem. The application of RLS$_o$ in graph mining is also briefly demonstrated.