Find k cliques in a graph

In Julia, there are multiple ways to find k cliques in a graph. In this article, we will explore three different approaches to solve this problem.

Approach 1: Using the LightGraphs package

The LightGraphs package in Julia provides a set of tools for working with graphs. To find k cliques in a graph using this package, we can follow these steps:

# Step 1: Install the LightGraphs package
using Pkg
Pkg.add("LightGraphs")

# Step 2: Import the necessary modules
using LightGraphs

# Step 3: Create a graph
graph = SimpleGraph(5)
add_edge!(graph, 1, 2)
add_edge!(graph, 1, 3)
add_edge!(graph, 2, 3)
add_edge!(graph, 3, 4)
add_edge!(graph, 4, 5)

# Step 4: Find k cliques
k = 3
cliques = maximal_cliques(graph, k)

# Step 5: Print the cliques
for clique in cliques
    println(clique)
end

This approach uses the LightGraphs package to create a graph and find k cliques. It is a straightforward and efficient solution.

Approach 2: Using the Graphs.jl package

Another option is to use the Graphs.jl package, which provides a high-level interface for working with graphs. Here’s how we can find k cliques using this package:

# Step 1: Install the Graphs.jl package
using Pkg
Pkg.add("Graphs")

# Step 2: Import the necessary modules
using Graphs

# Step 3: Create a graph
graph = SimpleGraph(5)
add_edge!(graph, 1, 2)
add_edge!(graph, 1, 3)
add_edge!(graph, 2, 3)
add_edge!(graph, 3, 4)
add_edge!(graph, 4, 5)

# Step 4: Find k cliques
k = 3
cliques = findcliques(graph, k)

# Step 5: Print the cliques
for clique in cliques
    println(clique)
end

This approach uses the Graphs.jl package to create a graph and find k cliques. It provides a high-level interface and is suitable for more complex graph operations.

Approach 3: Implementing a custom algorithm

If you prefer a more customized solution, you can implement your own algorithm to find k cliques in a graph. Here’s an example of how you can do it:

# Step 1: Create a graph representation
graph = Dict(1 => [2, 3], 2 => [1, 3], 3 => [1, 2, 4], 4 => [3, 5], 5 => [4])

# Step 2: Find k cliques
function find_k_cliques(graph, k, current_clique=[], cliques=[])
    if length(current_clique) == k
        push!(cliques, current_clique)
        return
    end
    
    for node in keys(graph)
        if all(neighbor in current_clique for neighbor in graph[node])
            find_k_cliques(graph, k, [current_clique..., node], cliques)
        end
    end
end

cliques = []
find_k_cliques(graph, 3, [], cliques)

# Step 3: Print the cliques
for clique in cliques
    println(clique)
end

This approach allows you to customize the algorithm to fit your specific requirements. However, it may require more effort to implement and optimize.

After comparing the three options, the best approach depends on your specific needs. If you prefer a simple and efficient solution, Approach 1 using the LightGraphs package is recommended. If you need more advanced graph operations, Approach 2 using the Graphs.jl package is a good choice. If you want full control and customization, Approach 3 with a custom algorithm is the way to go.