Note

The Approximate Densest Subgraph problem involves finding a subgraph of a given graph that has the highest density, where density is typically defined as the number of edges divided by the number of vertices in the subgraph. Finding the exact densest subgraph is computationally expensive, so approximate solutions are often sought.

Here’s a high-level outline of how an approximate algorithm for this problem might be implemented:

  1. Initialization: Start with all vertices of the graph and no edges.
  2. Iterative Removal: In each iteration, remove the vertex with the lowest degree (i.e., the least number of edges connected to it).
  3. Updating Density: After each removal, calculate the density of the remaining graph.
  4. Tracking the Best Subgraph: Keep track of the subgraph that has the highest density so far.
  5. Termination: The algorithm terminates when all vertices have been considered for removal. The subgraph with the highest density noted during the process is the output.

Code

#include <iostream>
#include <limits>
#include <vector>

float calculateDensity(const std::vector<int> &row_ptr,
                       const std::vector<int> &col_idx,
                       const std::vector<bool> &active_nodes) {
    int edge_count = 0;
    int active_node_count = 0;

    for (int i = 0; i < active_nodes.size(); i++) {
        if (active_nodes[i]) {
            active_node_count++;
            edge_count += row_ptr[i + 1] - row_ptr[i];
        }
    }
    return static_cast<float>(edge_count) / active_node_count;
}

std::vector<bool> approximateDensestSubgraph(const std::vector<int> &row_ptr,
                                             const std::vector<int> &col_idx) {
    const auto num_nodes = row_ptr.size() - 1;
    std::vector<bool> active_nodes(num_nodes, true);
    std::vector<bool> best_subgraph = active_nodes;
    float highest_density = 0.0;

    for (int i = 0; i < num_nodes; i++) {
				// **Updating Density**
        auto current_density = calculateDensity(row_ptr, col_idx, active_nodes);
        if (current_density > highest_density) {
            highest_density = current_density;
            // **Tracking the Best Subgraph**
						best_subgraph = active_nodes;
        }
				// **Iterative Removal**
        int min_degree = std::numeric_limits<int>::max();
        int min_degree_node = -1;
        for (int j = 0; j < num_nodes; j++) {
            if (active_nodes[j]) {
                auto degree = row_ptr[j + 1] - row_ptr[j];
                if (degree < min_degree) {
                    min_degree = degree;
                    min_degree_node = j;
                }
            }
        }
        if (min_degree_node != -1) {
            active_nodes[min_degree_node] = false;
        }
    }
    return best_subgraph;
}

int main() {
    std::vector<int> row_ptr = {0, 2, 4, 6};
    std::vector<int> col_idx = {1, 2, 0, 2, 0, 1};
    auto best_subgraph = approximateDensestSubgraph(row_ptr, col_idx);

    std::cout << "nodes in the densest subgraph:" << std::endl;
    for (size_t i = 0; i < best_subgraph.size(); ++i) {
        if (best_subgraph[i]) {
            std::cout << "node " << i << std::endl;
        }
    }

    return 0;
}