# Design and Analysis Max Cliques

In an undirected graph, a clique is a complete sub-graph of the given graph. Complete sub-graph means, all the vertices of this sub-graph is connected to all other vertices of this sub-graph.

The Max-Clique problem is the computational problem of finding maximum clique of the graph. Max clique is used in many real-world problems.

Let us consider a social networking application, where vertices represent people’s profile and the edges represent mutual acquaintance in a graph. In this graph, a clique represents a subset of people who all know each other.

To find a maximum clique, one can systematically inspect all subsets, but this sort of brute-force search is too time-consuming for networks comprising more than a few dozen vertices.

## Max-Clique Algorithm

The algorithm to find the maximum clique of a graph is relatively simple. The steps to the procedure are given below −

Step 1: Take a graph as an input to the algorithm with a non-empty set of vertices and edges.

Step 2: Create an output set and add the edges into it if they form a clique of the graph.

Step 3: Repeat Step 2 iteratively until all the vertices of the graph are checked, and the list does not form a clique further.

Step 4: Then the output set is backtracked to check which clique has the maximum edges in it.

### Pseudocode

Algorithm: Max-Clique (G, n, k)
S := ф
for i = 1 to k do
t := choice (1…n)
if t є S then
return failure
S := S U t
for all pairs (i, j) such that i є S and j є S and i ≠ j do
if (i, j) is not a edge of the graph then
return failure
return success


### Analysis

Max-Clique problem is a non-deterministic algorithm. In this algorithm, first we try to determine a set of k distinct vertices and then we try to test whether these vertices form a complete graph.

There is no polynomial time deterministic algorithm to solve this problem. This problem is NP-Complete.

Example

Take a look at the following graph. Here, the sub-graph containing vertices 2, 3, 4 and 6 forms a complete graph. Hence, this sub-graph is a clique. As this is the maximum complete sub-graph of the provided graph, it’s a 4-Clique.  