A Short Review on Novel Approaches for Maximum Clique Problem: from Classical algorithms to Graph Neural Networks and Quantum algorithms

TL;DR

This manuscript provides a comprehensive review of the Maximum Clique Problem, a computational problem that involves finding subsets of vertices in a graph that are all pairwise adjacent to each other and includes a review of recent developments in graph neural networks and quantum algorithms.

Abstract

This manuscript provides a comprehensive review of the Maximum Clique Problem, a computational problem that involves finding subsets of vertices in a graph that are all pairwise adjacent to each other. The manuscript covers in a simple way classical algorithms for solving the problem and includes a review of recent developments in graph neural networks and quantum algorithms. The review concludes with benchmarks for testing classical as well as new learning, and quantum algorithms.

Figures (1)

• Figure 1: An illustrative example of cliques within an undirected graph, showcasing the interconnected nodes and the formation of tightly-knit groups. In black the size of a clique of order $1$, in grey of order $2$, in yellow of order $3$, in orange of order $4$, in red of order $5$, in green of order $6$, and in blue of order $7$. This visualization emphasizes the concept of cliques as subsets of vertices within a graph, all of which are adjacent to each other, illustrating the density of connections that define these clusters. The maximum clique in the graph is the set of blue nodes.