Java Program to Find a Clique by Using the Technique of the Most Dense Subgraph

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A clique is a subset of vertices of a graph such that every two distinct vertices in the clique are adjacent. Finding a clique from a graph is an important problem in graph theory and many algorithms have been proposed to solve this problem. The most popular algorithm for finding a clique is the technique of the most dense subgraph. This technique is based on the fact that a clique is the most dense subgraph of the graph. In this article, we will discuss a Java program to find a clique by using the technique of the most dense subgraph.

Examples

Example 1:

Let’s take a graph G = (V, E) with the following adjacency matrix:

V = {v1, v2, v3, v4}

E = {(v1, v2), (v2, v3), (v3, v4), (v1, v4)}

Adjacency Matrix:
[[0, 1, 0, 1],
[1, 0, 1, 0],
[0, 1, 0, 1],
[1, 0, 1, 0]]

The most dense subgraph of the graph G is the clique {v1, v2, v3, v4}.

Example 2:

Let’s take another graph G = (V, E) with the following adjacency matrix:

V = {v1, v2, v3, v4, v5}

E = {(v1, v2), (v2, v3), (v3, v4), (v4, v5), (v1, v5)}

Adjacency Matrix:

[[0, 1, 0, 0, 1],
[1, 0, 1, 0, 0],
[0, 1, 0, 1, 0],
[0, 0, 1, 0, 1],
[1, 0, 0, 1, 0]]
The most dense subgraph of the graph G is the clique {v1, v2, v3, v4}.

Approach

The approach used in the program is based on the technique of the most dense subgraph. The algorithm works as follows:

Initialize an array of size equal to the number of vertices in the graph.
For each vertex, count the number of edges connected to it.
Find the vertex with the maximum number of edges and mark it as the first vertex of the clique.
For each of the remaining vertices in the graph, count the number of edges connected to both the first vertex and the current vertex.
If the number of edges is equal to the number of vertices in the clique, then add the current vertex to the clique.
Repeat Steps 4 and 5 until all the vertices have been checked.

Below is the implementation of the above approach.

Implementation

Java

// Java Program for the above approach
import java.util.ArrayList;
import java.util.List;
  
public class CliqueFinder {
  
    static int[][] adjMatrix;
  
    static int cliqueSize;
  
    static List<Integer> clique;
  
    // Function to initialize the adjacency matrix
    static void init(int[][] adjMatrix)
    {
        clique = new ArrayList<Integer>();
        for (int i = 0; i < adjMatrix.length; i++) {
            int count = 0;
            for (int j = 0; j < adjMatrix.length; j++) {
                if (adjMatrix[i][j] == 1)
                    count++;
            }
            if (count > cliqueSize) {
                cliqueSize = count;
                clique.clear();
                clique.add(i);
            }
        }
    }
  
    // Function to find the clique
    static void findClique(int[][] adjMatrix)
    {
        for (int i = 0; i < adjMatrix.length; i++) {
            if (!clique.contains(i)) {
                int count = 0;
                for (int j = 0; j < clique.size(); j++) {
                    if (adjMatrix[i][clique.get(j)] == 1)
                        count++;
                }
                if (count == cliqueSize) {
                    clique.add(i);
                }
            }
        }
    }
  
    // Driver code
    public static void main(String[] args)
    {
        int[][] adjMatrix = { { 0, 1, 0, 1 },
                              { 1, 0, 1, 0 },
                              { 0, 1, 0, 1 },
                              { 1, 0, 1, 0 } };
  
        init(adjMatrix);
        findClique(adjMatrix);
  
        // Print the clique
        System.out.println("The clique is: ");
        for (int i = 0; i < clique.size(); i++) {
            System.out.print(clique.get(i) + " ");
        }
    }
}