# Compute the Maximum Clique in a Graph

## Introduction to Maximum Clique Problem

The Maximum Clique Problem (MCP) is an NP-hard combinatorial optimization problem in graph theory. It involves finding the largest complete subgraph, or a *clique*, within a given graph. A clique is a subset of vertices in which every pair of vertices is directly connected by an edge. In other words, a clique is a group of vertices that are all connected to one another.

The Maximum Clique Problem has various applications in real-world scenarios, such as social network analysis, bioinformatics, and network security. In this tutorial, we will explore a real-world example involving social networks, discuss a formal definition of the problem, and provide a step-by-step solution using the Bron-Kerbosch algorithm.

## Real-world Examples and Scenarios

### Social Network Analysis

In the context of social networks, vertices represent users, and edges represent connections between users. A clique in a social network could represent a group of friends who all know each other. The Maximum Clique Problem can be used to identify the largest group of friends who all know each other or to detect communities within a social network.

### Bioinformatics

In bioinformatics, researchers often represent protein-protein interactions using graphs. Vertices represent proteins, and edges represent interactions between proteins. A clique in a protein-protein interaction network could represent a group of proteins that all interact with each other. The Maximum Clique Problem can help identify potential protein complexes or functional modules within a cell.

### Network Security

In network security, vertices represent devices in a network, and edges represent connections between devices. A clique in a network could represent a group of devices that are all connected to each other. The Maximum Clique Problem can be used to identify groups of devices that may represent a security risk or to detect potential vulnerabilities in a network.

## Real-World Scenario: Social Network Analysis

Let's consider a scenario where we have a social network of users, and we want to identify the largest group of friends who all know each other. This is an instance of the Maximum Clique Problem.

### Problem Statement with a Formal Definition

Given an undirected graph G = (V, E), where V is the set of vertices representing users and E is the set of edges representing connections between users, the Maximum Clique Problem (MCP) is to find a clique C in G such that the size of C is maximized.

### Solution to the Problem: Bron-Kerbosch Algorithm

The Bron-Kerbosch algorithm is a recursive backtracking algorithm that finds all maximal cliques in an undirected graph. It is particularly efficient for sparse graphs and can be used to solve the Maximum Clique Problem by finding all maximal cliques and selecting the largest one.

Here's a high-level description of the Bron-Kerbosch algorithm:

- Initialize three sets: R (the current clique), P (the candidate vertices), and X (the excluded vertices).
- If both P and X are empty, the current clique R is maximal, so report it.
- For each vertex v in P: a. Recursively call the algorithm with R ⋃ {v}, P ⋂ N(v), and X ⋂ N(v), where N(v) is the set of neighbors of v. b. Remove v from P and add it to X.

### Step-by-Step Solution with the Social Network Scenario

Let's go through the Bron-Kerbosch algorithm step by step using a sample social network graph.

- Initialize R = {}, P = {1, 2, 3, 4, 5}, and X = {}.
- Since P and X are not empty, we proceed to step 3.
- For each vertex in P: a. Recursive calls:

- Call with R = {1}, P = {2, 3, 4}, X = {}.
- Call with R = {2}, P = {3, 4}, X = {1}.
- Call with R = {3}, P = {4}, X = {1, 2}.
- Call with R = {4}, P = {}, X = {1, 2, 3}. b. Update P and X:
- P = {2, 3, 4, 5}, X = {1}.
- P = {3, 4, 5}, X = {1, 2}.
- P = {4, 5}, X = {1, 2, 3}.
- P = {5}, X = {1, 2, 3, 4}.

The algorithm finds the following maximal cliques: {1, 2, 3}, {1, 4}, {2, 4}, and {3, 4}. The largest clique is {1, 2, 3}, so the maximum clique size is 3.

### Code Example

Here's a Python implementation of the Bron-Kerbosch algorithm:

```
def bron_kerbosch(r, p, x, graph):
if not p and not x:
print(r)
return
for v in p.copy():
bron_kerbosch(r | {v}, p & graph[v], x & graph[v], graph)
p.remove(v)
x.add(v)
graph = {
1: {2, 3, 4},
2: {1, 3, 4},
3: {1, 2, 4},
4: {1, 2, 3},
5: {}
}
bron_kerbosch(set(), set(graph.keys()), set(), graph)
```

### Intuitions and Analogies

The Bron-Kerbosch algorithm is like searching through a tree where each node represents a partial solution, and each level of the tree adds a new vertex to the clique. The algorithm traverses the tree using a depth-first search and backtracks when it reaches a dead end.

### Solving Other Real-World Problems

The Bron-Kerbosch algorithm can be applied to other real-world problems involving the Maximum Clique Problem, such as identifying protein complexes in bioinformatics or detecting network vulnerabilities in network security. The same algorithm can be used, with the input graph representing the problem-specific domain.

In summary, we have explored the Maximum Clique Problem, discussed its real-world applications, and provided a step-by-step solution using the Bron-Kerbosch algorithm. This algorithm can be applied to various scenarios, such as social network analysis, bioinformatics, and network security, to efficiently solve the Maximum Clique Problem.