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

A clique in a graph is a subset of vertices where every pair of vertices is connected by an edge. Finding the largest clique is computationally challenging, but we can determine the minimum size of the largest clique using a binary search approach with edge counting.

The problem asks: given the number of nodes and edges in a graph, what is the minimum possible size of the largest clique?

Understanding the Problem

If we have a graph with n nodes and e edges, we need to find the smallest value k such that any graph with these parameters must contain a clique of size at least k.

Algorithm Approach

We use binary search to find the minimum clique size. The helper function calculates the maximum number of edges possible when partitioning nodes to avoid a clique of size p.

Helper Function Logic

To avoid a clique of size p, we partition n nodes into p-1 groups. The maximum edges occur when groups are as equal as possible ?

def helper(x, y):
    # Partition x nodes into y groups to avoid clique of size y+1
    ga = x % y  # remainder nodes
    gb = y - ga  # groups with floor(x/y) nodes
    sa = x // y + 1  # size of larger groups
    sb = x // y      # size of smaller groups
    
    # Calculate maximum edges within groups
    edges_large = ga * (ga - 1) * sa * sa // 2
    edges_small = gb * (gb - 1) * sb * sb // 2
    edges_between = ga * gb * sa * sb
    
    return edges_between + edges_large + edges_small

def solve(nodes, edges):
    i = 1
    j = nodes + 1
    
    # Binary search for minimum clique size
    while i + 1 < j:
        p = i + (j - i) // 2
        max_edges_without_clique_p = helper(nodes, p)
        
        if max_edges_without_clique_p < edges:
            i = p  # Need larger clique
        else:
            j = p  # Can avoid clique of size p
    
    return j

# Test with the example
result = solve(4, 4)
print(f"Minimum largest clique size: {result}")

Minimum largest clique size: 2

How It Works

The algorithm uses the Turán's theorem principle: to avoid a clique of size k, partition the graph into k-1 independent sets. The maximum edges without forming a clique of size p is calculated by the helper function.

Step-by-Step Execution

def solve_with_steps(nodes, edges):
    print(f"Finding minimum clique size for {nodes} nodes, {edges} edges")
    i = 1
    j = nodes + 1
    step = 1
    
    while i + 1 < j:
        p = i + (j - i) // 2
        max_edges = helper(nodes, p)
        print(f"Step {step}: Testing clique size {p}")
        print(f"  Max edges without clique-{p}: {max_edges}")
        
        if max_edges < edges:
            print(f"  {max_edges} < {edges}, need larger clique")
            i = p
        else:
            print(f"  {max_edges} >= {edges}, can avoid clique-{p}")
            j = p
        
        step += 1
    
    return j

result = solve_with_steps(4, 4)
print(f"\nFinal answer: {result}")

Finding minimum clique size for 4 nodes, 4 edges
Step 1: Testing clique size 2
  Max edges without clique-2: 4
  4 >= 4, can avoid clique-2
Step 2: Testing clique size 1
  Max edges without clique-1: 0
  0 < 4, need larger clique

Final answer: 2

Key Points

• The algorithm finds the minimum guaranteed clique size for any graph with given nodes and edges
• Uses binary search with Turán's theorem for efficiency
• Time complexity: O(log n) where n is the number of nodes
• Helper function calculates maximum edges in a clique-free partition

Conclusion

This approach efficiently determines the minimum size of the largest clique using binary search and graph theory. The helper function ensures we find the theoretical minimum based on edge distribution across partitioned node groups.