Solving Network Flow Problems with Algorithms Like Ford-Fulkerson

Network flow problems are crucial in areas like transportation, communication networks, water distribution, and computer networking. These problems involve determining the maximum flow that can be sent from a source node to a sink node in a network with limited capacity on edges.

One of the most fundamental algorithms for solving network flow problems is the Ford-Fulkerson Algorithm, which finds the maximum flow in a flow network using the concept of augmenting paths.

In this article, we will explore:

• What is a flow network?
• How the Ford-Fulkerson algorithm works.
• Implementation in C++ with explanation.
• Real-world applications.

1. Understanding Flow Networks

A flow network is a directed graph where:

• Each edge has a capacity that limits how much flow can pass through it.
• There is a source node (s) where flow originates.
• There is a sink node (t) where flow is collected.
• The goal is to maximize the flow from the source to the sink while ensuring flow constraints are met.

Key Terms in Network Flow

✅ Capacity (c(u, v)) – The maximum amount of flow an edge (u, v) can carry.
✅ Flow (f(u, v)) – The actual amount of flow passing through an edge.
✅ Residual Capacity – The remaining capacity after accounting for flow: residual capacity = c(u,v) − f(u,v)

✅ Augmenting Path – A path from the source to the sink with available residual capacity.
✅ Residual Graph – A modified graph that keeps track of unused capacities and allows backflows.

2. Ford-Fulkerson Algorithm: Maximum Flow Computation

How Ford-Fulkerson Works

Ford-Fulkerson is a greedy, iterative approach that finds augmenting paths and pushes as much flow as possible through them until no more augmenting paths exist.

Algorithm Steps

1. Initialize the flow on all edges to 0.
2. Find an augmenting path from source to sink in the residual graph using DFS or BFS.
3. Find the minimum residual capacity (bottleneck) along the path.
4. Augment the flow along the path by adding the bottleneck value.
5. Update the residual graph to reflect the changes.
6. Repeat until no more augmenting paths exist.

3. C++ Implementation of Ford-Fulkerson Algorithm

Adjacency Matrix Implementation Using DFS

#include <iostream>
#include <vector>
#include <climits>
#include <cstring>

using namespace std;

#define V 6  // Number of vertices in the graph

// Perform Depth-First Search (DFS) to find an augmenting path
bool dfs(int rGraph[V][V], int u, int t, vector<int>& parent, vector<bool>& visited) {
    visited[u] = true;
    
    if (u == t) return true; // If we reach the sink, path found
    
    for (int v = 0; v < V; v++) {
        if (!visited[v] && rGraph[u][v] > 0) { // Check residual capacity
            parent[v] = u;
            if (dfs(rGraph, v, t, parent, visited))
                return true;
        }
    }
    return false;
}

// Ford-Fulkerson Algorithm for Maximum Flow
int fordFulkerson(int graph[V][V], int s, int t) {
    int rGraph[V][V]; // Residual graph
    memcpy(rGraph, graph, sizeof(rGraph));
    
    vector<int> parent(V); // Store path
    int maxFlow = 0;

    while (true) {
        vector<bool> visited(V, false);
        
        // Find an augmenting path using DFS
        if (!dfs(rGraph, s, t, parent, visited))
            break; // No more augmenting paths
        
        // Find the bottleneck (minimum residual capacity)
        int pathFlow = INT_MAX;
        for (int v = t; v != s; v = parent[v]) {
            int u = parent[v];
            pathFlow = min(pathFlow, rGraph[u][v]);
        }
        
        // Update residual graph
        for (int v = t; v != s; v = parent[v]) {
            int u = parent[v];
            rGraph[u][v] -= pathFlow;
            rGraph[v][u] += pathFlow; // Reverse flow
        }

        maxFlow += pathFlow; // Add path flow to total max flow
    }

    return maxFlow;
}

// Driver Code
int main() {
    int graph[V][V] = {
        {0, 16, 13, 0, 0, 0},
        {0, 0, 10, 12, 0, 0},
        {0, 4, 0, 0, 14, 0},
        {0, 0, 9, 0, 0, 20},
        {0, 0, 0, 7, 0, 4},
        {0, 0, 0, 0, 0, 0}
    };

    int maxFlow = fordFulkerson(graph, 0, 5);
    cout << "The Maximum Flow is: " << maxFlow << endl;
    
    return 0;
}

4. Time Complexity Analysis

• Finding an augmenting path (DFS/BFS) takes O(V + E).
• Since flow increases in each iteration, at most O(F) augmenting paths exist, where F is the maximum flow.
• Worst-case time complexity: O(E⋅F)
  This can be slow for large graphs with high capacity values.

Optimizations

1. Edmonds-Karp Algorithm (BFS-based Ford-Fulkerson) → O(VE²).
2. Push-Relabel Algorithm → Faster for large graphs.

5. Real-World Applications of Network Flow Algorithms

🚀 1. Transportation and Logistics

• Finding the maximum capacity in road and rail networks.
• Optimizing airline and cargo shipments.

🌍 2. Internet and Data Networks

• Maximizing bandwidth usage in computer networks.
• Efficient data packet routing.

⚡ 3. Power Grid Optimization

• Balancing electrical load between stations.

🏥 4. Healthcare Resource Allocation

• Assigning hospitals to patients based on available resources.

📊 5. Bipartite Matching Problems

• Job assignments (matching workers to tasks).
• University admissions (assigning students to schools).

6. Conclusion

The Ford-Fulkerson Algorithm is a powerful technique for solving maximum flow problems in networks. It finds an augmenting path and iteratively increases flow until no more flow can be added.

🔹 Key Takeaways:

• Uses greedy approach with augmenting paths.
• Time complexity: O(E * F) (not always efficient for large graphs).
• Can be optimized using BFS (Edmonds-Karp) or push-relabel techniques.