Program to find out the minimum size of the largest clique in a graph (Python)

Suppose we are given a graph and are asked to find out the minimum size of the largest clique in the graph. A clique of a graph is a subset of a graph where every pair of vertices are adjacent, i.e. there exists an edge between every pair of vertices. Finding the largest clique in a graph is not possible in polynomial time, so given the number of nodes and edges of a small graph we shall have to find out the largest clique in it.

So, if the input is like nodes = 4, edges =4; then the output will be 2.

In the graph above, the maximum size of a clique is 2.

To solve this, we will follow these steps −

• Define a function helper() . This will take x, y
  • ga := x mod y
  • gb := y - ga
  • sa := quotient of value of(x / y) + 1
  • sb := quotient of value of(x / y)
  • return ga * gb * sa * sb + ga *(ga - 1) * sa * sa / 2 + gb * (gb - 1) * sb * sb / 2
• i := 1
• j := nodes + 1
• while i + 1 < j, do
  • p := i + floor value of((j - i) / 2)
  • k := helper(nodes, p)
  • if k < edges, then
    • i := p
  • otherwise,
    • j := p
• return j

Example

Let us see the following implementation to get better understanding −

import math def helper(x, y):     ga = x % y     gb = y - ga     sa = x // y + 1     sb = x // y     return ga * gb * sa * sb + ga * (ga - 1) * sa * sa // 2 + gb * (gb - 1) * sb * sb // 2 def solve(nodes, edges):     i = 1     j = nodes + 1     while i + 1 < j:         p = i + (j - i) // 2         k = helper(nodes, p)         if k < edges:             i = p         else:             j = p     return j print(solve(4, 4)) 

Input

4,4

Output

2