Graph Theory Questions Medium
The maximum clique problem in graph theory refers to the task of finding the largest complete subgraph, also known as a clique, within a given graph. A clique is a subset of vertices in a graph where every pair of vertices is connected by an edge. The maximum clique problem aims to determine the clique with the maximum number of vertices in a graph.
Formally, given an undirected graph G = (V, E), the maximum clique problem seeks to find a subset C ⊆ V such that every pair of vertices in C is connected by an edge, and C is not a proper subset of any other clique in G. The goal is to find the largest possible clique in the graph.
Solving the maximum clique problem is known to be an NP-hard problem, meaning that there is no known efficient algorithm that can solve it for all instances in polynomial time. Therefore, various approximation algorithms and heuristics are commonly used to find reasonably large cliques in practice.
The maximum clique problem has numerous applications in various fields, including computer science, social network analysis, bioinformatics, and operations research. It is often used to model and solve real-world problems that involve finding cohesive groups or communities within a network.