Find minimum s-t cut in a flow network

• Difficulty Level : Hard
• Last Updated : 10 Oct, 2020

In a flow network, an s-t cut is a cut that requires the source ‘s’ and the sink ‘t’ to be in different subsets, and it consists of edges going from the source’s side to the sink’s side. The capacity of an s-t cut is defined by the sum of the capacity of each edge in the cut-set. (Source: Wiki)
The problem discussed here is to find minimum capacity s-t cut of the given network. Expected output is all edges of the minimum cut.

For example, in the following flow network, example s-t cuts are {{0 ,1}, {0, 2}}, {{0, 2}, {1, 2}, {1, 3}}, etc. The minimum s-t cut is {{1, 3}, {4, 3}, {4 5}} which has capacity as 12+7+4 = 23.

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution. We strongly recommend to read the below post first.
Ford-Fulkerson Algorithm for Maximum Flow Problem

Minimum Cut and Maximum Flow
Like Maximum Bipartite Matching, this is another problem which can solved using Ford-Fulkerson Algorithm. This is based on max-flow min-cut theorem.

The max-flow min-cut theorem states that in a flow network, the amount of maximum flow is equal to capacity of the minimum cut. See CLRS book for proof of this theorem.

From Ford-Fulkerson, we get capacity of minimum cut. How to print all edges that form the minimum cut? The idea is to use residual graph.

Following are steps to print all edges of the minimum cut.

1) Run Ford-Fulkerson algorithm and consider the final residual graph.

2) Find the set of vertices that are reachable from the source in the residual graph.

3) All edges which are from a reachable vertex to non-reachable vertex are minimum cut edges. Print all such edges.

Following is the implementation of the above approach.

C++

// C++ program for finding minimum cut using Ford-Fulkerson
#include <iostream>
#include <limits.h>
#include <string.h>
#include <queue>
using namespace std;

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

/* Returns true if there is a path from source 's' to sink 't' in
residual graph. Also fills parent[] to store the path */
int bfs(int rGraph[V][V], int s, int t, int parent[])
{
// Create a visited array and mark all vertices as not visited
bool visited[V];
memset(visited, 0, sizeof(visited));

// Create a queue, enqueue source vertex and mark source vertex
// as visited
queue <int> q;
q.push(s);
visited[s] = true;
parent[s] = -1;

// Standard BFS Loop
while (!q.empty())
{
int u = q.front();
q.pop();

for (int v=0; v<V; v++)
{
if (visited[v]==false && rGraph[u][v] > 0)
{
q.push(v);
parent[v] = u;
visited[v] = true;
}
}
}

// If we reached sink in BFS starting from source, then return
// true, else false
return (visited[t] == true);
}

// A DFS based function to find all reachable vertices from s.  The function
// marks visited[i] as true if i is reachable from s.  The initial values in
// visited[] must be false. We can also use BFS to find reachable vertices
void dfs(int rGraph[V][V], int s, bool visited[])
{
visited[s] = true;
for (int i = 0; i < V; i++)
if (rGraph[s][i] && !visited[i])
dfs(rGraph, i, visited);
}

// Prints the minimum s-t cut
void minCut(int graph[V][V], int s, int t)
{
int u, v;

// Create a residual graph and fill the residual graph with
// given capacities in the original graph as residual capacities
// in residual graph
int rGraph[V][V]; // rGraph[i][j] indicates residual capacity of edge i-j
for (u = 0; u < V; u++)
for (v = 0; v < V; v++)
rGraph[u][v] = graph[u][v];

int parent[V];  // This array is filled by BFS and to store path

// Augment the flow while there is a path from source to sink
while (bfs(rGraph, s, t, parent))
{
// Find minimum residual capacity of the edhes along the
// path filled by BFS. Or we can say find the maximum flow
// through the path found.
int path_flow = INT_MAX;
for (v=t; v!=s; v=parent[v])
{
u = parent[v];
path_flow = min(path_flow, rGraph[u][v]);
}

// update residual capacities of the edges and reverse edges
// along the path
for (v=t; v != s; v=parent[v])
{
u = parent[v];
rGraph[u][v] -= path_flow;
rGraph[v][u] += path_flow;
}
}

// Flow is maximum now, find vertices reachable from s
bool visited[V];
memset(visited, false, sizeof(visited));
dfs(rGraph, s, visited);

// Print all edges that are from a reachable vertex to
// non-reachable vertex in the original graph
for (int i = 0; i < V; i++)
for (int j = 0; j < V; j++)
if (visited[i] && !visited[j] && graph[i][j])
cout << i << " - " << j << endl;

return;
}

// Driver program to test above functions
int main()
{
// Let us create a graph shown in the above example
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}
};

minCut(graph, 0, 5);

return 0;
}

Java

// Java program for finding min-cut in the given graph
import java.util.Queue;

public class Graph {

// Returns true if there is a path
// from source 's' to sink 't' in residual
// graph. Also fills parent[] to store the path
private static boolean bfs(int[][] rGraph, int s,
int t, int[] parent) {

// Create a visited array and mark
// all vertices as not visited
boolean[] visited = new boolean[rGraph.length];

// Create a queue, enqueue source vertex
// and mark source vertex as visited
visited[s] = true;
parent[s] = -1;

// Standard BFS Loop
while (!q.isEmpty()) {
int v = q.poll();
for (int i = 0; i < rGraph.length; i++) {
if (rGraph[v][i] > 0 && !visited[i]) {
q.offer(i);
visited[i] = true;
parent[i] = v;
}
}
}

// If we reached sink in BFS starting
// from source, then return true, else false
return (visited[t] == true);
}

// A DFS based function to find all reachable
// vertices from s. The function marks visited[i]
// as true if i is reachable from s. The initial
// values in visited[] must be false. We can also
// use BFS to find reachable vertices
private static void dfs(int[][] rGraph, int s,
boolean[] visited) {
visited[s] = true;
for (int i = 0; i < rGraph.length; i++) {
if (rGraph[s][i] > 0 && !visited[i]) {
dfs(rGraph, i, visited);
}
}
}

// Prints the minimum s-t cut
private static void minCut(int[][] graph, int s, int t) {
int u,v;

// Create a residual graph and fill the residual
// graph with given capacities in the original
// graph as residual capacities in residual graph
// rGraph[i][j] indicates residual capacity of edge i-j
int[][] rGraph = new int[graph.length][graph.length];
for (int i = 0; i < graph.length; i++) {
for (int j = 0; j < graph.length; j++) {
rGraph[i][j] = graph[i][j];
}
}

// This array is filled by BFS and to store path
int[] parent = new int[graph.length];

// Augment the flow while tere is path from source to sink
while (bfs(rGraph, s, t, parent)) {

// Find minimum residual capacity of the edhes
// along the path filled by BFS. Or we can say
// find the maximum flow through the path found.
int pathFlow = Integer.MAX_VALUE;
for (v = t; v != s; v = parent[v]) {
u = parent[v];
pathFlow = Math.min(pathFlow, rGraph[u][v]);
}

// update residual capacities of the edges and
// reverse edges along the path
for (v = t; v != s; v = parent[v]) {
u = parent[v];
rGraph[u][v] = rGraph[u][v] - pathFlow;
rGraph[v][u] = rGraph[v][u] + pathFlow;
}
}

// Flow is maximum now, find vertices reachable from s
boolean[] isVisited = new boolean[graph.length];
dfs(rGraph, s, isVisited);

// Print all edges that are from a reachable vertex to
// non-reachable vertex in the original graph
for (int i = 0; i < graph.length; i++) {
for (int j = 0; j < graph.length; j++) {
if (graph[i][j] > 0 && isVisited[i] && !isVisited[j]) {
System.out.println(i + " - " + j);
}
}
}
}

//Driver Program
public static void main(String args[]) {

// Let us create a graph shown in the above example
int graph[][] = { {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}
};
minCut(graph, 0, 5);
}
}
// This code is contributed by Himanshu Shekhar

Python

# Python program for finding min-cut in the given graph
# Complexity : (E*(V^3))
# Total augmenting path = VE and BFS
# with adj matrix takes :V^2 times

from collections import defaultdict

# This class represents a directed graph
class Graph:

def __init__(self,graph):
self.graph = graph # residual graph
self.org_graph = [i[:] for i in graph]
self. ROW = len(graph)
self.COL = len(graph)

'''Returns true if there is a path from
source 's' to sink 't' in
residual graph. Also fills
parent[] to store the path '''
def BFS(self,s, t, parent):

# Mark all the vertices as not visited
visited =[False]*(self.ROW)

# Create a queue for BFS
queue=[]

# Mark the source node as visited and enqueue it
queue.append(s)
visited[s] = True

# Standard BFS Loop
while queue:

#Dequeue a vertex from queue and print it
u = queue.pop(0)

# Get all adjacent vertices of
# the dequeued vertex u
# If a adjacent has not been
# visited, then mark it
# visited and enqueue it
for ind, val in enumerate(self.graph[u]):
if visited[ind] == False and val > 0 :
queue.append(ind)
visited[ind] = True
parent[ind] = u

# If we reached sink in BFS starting
# from source, then return
# true, else false
return True if visited[t] else False

# Function for Depth first search
# Traversal of the graph
def dfs(self, graph,s,visited):
visited[s]=True
for i in range(len(graph)):
if graph[s][i]>0 and not visited[i]:
self.dfs(graph,i,visited)

# Returns the min-cut of the given graph
def minCut(self, source, sink):

# This array is filled by BFS and to store path
parent = [-1]*(self.ROW)

max_flow = 0 # There is no flow initially

# Augment the flow while there is path from source to sink
while self.BFS(source, sink, parent) :

# Find minimum residual capacity of the edges along the
# path filled by BFS. Or we can say find the maximum flow
# through the path found.
path_flow = float("Inf")
s = sink
while(s != source):
path_flow = min (path_flow, self.graph[parent[s]][s])
s = parent[s]

# Add path flow to overall flow
max_flow += path_flow

# update residual capacities of the edges and reverse edges
# along the path
v = sink
while(v != source):
u = parent[v]
self.graph[u][v] -= path_flow
self.graph[v][u] += path_flow
v = parent[v]

visited=len(self.graph)*[False]
self.dfs(self.graph,s,visited)

# print the edges which initially had weights
# but now have 0 weight
for i in range(self.ROW):
for j in range(self.COL):
if self.graph[i][j] == 0 and\
self.org_graph[i][j] > 0 and visited[i]:
print str(i) + " - " + str(j)

# Create a graph given in the above diagram
graph = [[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]]

g = Graph(graph)

source = 0; sink = 5

g.minCut(source, sink)

# This code is contributed by Neelam Yadav

C#

// C# program for finding min-cut in the given graph
using System;
using System.Collections.Generic;

class Graph
{

// Returns true if there is a path
// from source 's' to sink 't' in residual
// graph. Also fills parent[] to store the path
private static bool bfs(int[,] rGraph, int s,
int t, int[] parent)
{

// Create a visited array and mark
// all vertices as not visited
bool[] visited = new bool[rGraph.Length];

// Create a queue, enqueue source vertex
// and mark source vertex as visited
Queue<int> q = new Queue<int>();
q.Enqueue(s);
visited[s] = true;
parent[s] = -1;

// Standard BFS Loop
while (q.Count != 0)
{
int v = q.Dequeue();
for (int i = 0; i < rGraph.GetLength(0); i++)
{
if (rGraph[v,i] > 0 && !visited[i])
{
q.Enqueue(i);
visited[i] = true;
parent[i] = v;
}
}
}

// If we reached sink in BFS starting
// from source, then return true, else false
return (visited[t] == true);
}

// A DFS based function to find all reachable
// vertices from s. The function marks visited[i]
// as true if i is reachable from s. The initial
// values in visited[] must be false. We can also
// use BFS to find reachable vertices
private static void dfs(int[,] rGraph, int s,
bool[] visited)
{
visited[s] = true;
for (int i = 0; i < rGraph.GetLength(0); i++)
{
if (rGraph[s,i] > 0 && !visited[i])
{
dfs(rGraph, i, visited);
}
}
}

// Prints the minimum s-t cut
private static void minCut(int[,] graph, int s, int t)
{
int u, v;

// Create a residual graph and fill the residual
// graph with given capacities in the original
// graph as residual capacities in residual graph
// rGraph[i,j] indicates residual capacity of edge i-j
int[,] rGraph = new int[graph.Length,graph.Length];
for (int i = 0; i < graph.GetLength(0); i++)
{
for (int j = 0; j < graph.GetLength(1); j++)
{
rGraph[i, j] = graph[i, j];
}
}

// This array is filled by BFS and to store path
int[] parent = new int[graph.Length];

// Augment the flow while there is path
// from source to sink
while (bfs(rGraph, s, t, parent))
{

// Find minimum residual capacity of the edhes
// along the path filled by BFS. Or we can say
// find the maximum flow through the path found.
int pathFlow = int.MaxValue;
for (v = t; v != s; v = parent[v])
{
u = parent[v];
pathFlow = Math.Min(pathFlow, rGraph[u, v]);
}

// update residual capacities of the edges and
// reverse edges along the path
for (v = t; v != s; v = parent[v])
{
u = parent[v];
rGraph[u, v] = rGraph[u, v] - pathFlow;
rGraph[v, u] = rGraph[v, u] + pathFlow;
}
}

// Flow is maximum now, find vertices reachable from s
bool[] isVisited = new bool[graph.Length];
dfs(rGraph, s, isVisited);

// Print all edges that are from a reachable vertex to
// non-reachable vertex in the original graph
for (int i = 0; i < graph.GetLength(0); i++)
{
for (int j = 0; j < graph.GetLength(1); j++)
{
if (graph[i, j] > 0 &&
isVisited[i] && !isVisited[j])
{
Console.WriteLine(i + " - " + j);
}
}
}
}

// Driver Code
public static void Main(String []args)
{

// Let us create a graph shown
// in the above example
int [,]graph = {{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}};
minCut(graph, 0, 5);
}
}

// This code is contributed by PrinciRaj1992

Output:
1 - 3
4 - 3
4 - 5