Max-flow min-cut
Overview of the Max-flow problem
with sample code and example problem.
Georgi Stoyanov
Student at
Sofia University
http://backtrack-it.blogspot.com
Table of Contents
1.
Definition of the problem
2.
Where does it occur?
3.
Max-flow min-cut theorem
4.
Example
5.
Max-flow algorithm
6.
Run-time estimation
7.
Questions
2
Definition of the problem
Definition of the problem
 Maximum flow problems
 Finding feasible flow
 Through a single -source, -sink flow network
 Flow is maximum
 Many problems solved by Max-flow
 The problem is often present at
algorithmic competitions
The Max-flow algorithm
 Additional
definitions
 Edge capacity – maximum flow that can go
through the edge
 Residual edge capacity – maximum flow that
can pass after a certain amount has passed
 residualCapacity = edgeCapacity – alreadyPassedFlow
 Augmented path – path starting from source to
sink
 Only edges with residual capacity above zero
5
Where does it occur?
Where does it occur?
 In any kind of network with certain capacity
 Network of pipes – how much water can pass
through the pipe network per unit of time?
7
Where does it occur?
 Electricity network – how much electricity can
go through the grid?
8
Where does it occur?
 The internet network – how much traffic can go
through a local network or the internet?
9
Where does it occur?
 In other problems
 Matching problem
 Group of N guys and M girls
 Every girl/guy likes a certain amount of people
from the other group
 What is the maximum
number of couples, with
people who like each other?
10
Where does it occur?

Converting the matching problem to a max-flow
problem:
 We add an edge with capacity one for every couple
that is acceptable
 We add two bonus nodes – source and sink
 We connect the source with the first group and the
second group with the sink
11
Max-flow min-cut theorem
Max-flow min-cut theorem

The max-flow min-cut theorem states that in a flow
network, the maximum amount of flow passing from
the source to the sink is equal to the minimum capacity
that when removed in a specific way from the network
causes the situation that no flow can pass from the
source to the sink.
13
Example
Example
 Example
min( cf(A,D), cf(D,E), cf(E,G)) =
min( 3 – 0, 2 – 0, 1 – 0) =
min( 3, 2, 1) = 1
maxFlow = maxFlow + 1 = 1
15
Example
 Example
min( cf(A,D), cf(D,F), cf(F,G)) =
min( 3 – 1, 6 – 0, 9 – 0) =
min( 2, 6, 9) = 2
maxFlow = maxFlow + 2 = 3
16
Example
 Example
min( cf(A,B), cf(B,C), cf(C,D), cf(D,F), cf(F,G)) =
min( 3 – 0, 4 – 0, 1 – 0, 6 – 2, 9 - 2) =
min( 3, 4, 1, 4, 7) = 1
maxFlow = maxFlow + 1 = 4
17
Example
 The flow in the previous slide is not optimal!
 Reverting some of the flow through a different
path will achieve the optimal answer
 To do that for each directed edge (u, v) we will
add an imaginary reverse edge (v, u)
 The new edge shall
be used only if a certain
amount of flow has already passed through
the original edge!
18
Example
 Example
min( cf(A,B), cf(B,C), cf(C,E), cf(E,D), cf(D,F), cf(F,g) ) =
min( 3 – 1, 4 – 1, 2 – 0, 0 – -1, 6 – 3, 9 - 3) =
min( 2, 3, 2, 1, 3, 6 ) = 1
maxFlow = maxFlow + 1 = 5 (which is the final answer)
19
The Max-flow algorithm
The Max-flow algorithm
 The Edmonds-Karp algorithm
 Uses a graph structure
 Uses matrix of the capacities
 Uses matrix for the passed flow
21
The Max-flow algorithm
 The Edmonds-Karp algorithm
 Uses breadth-first search on each iteration to
find a path from the source to the sink
 Uses parent table to store the path
 Uses path capacity table to store the value of
the maximum flow to a node in the path
22
The Max-flow algorithm initialization
#include<cstdio>
#include<queue>
#include<cstring>
#include<vector>
#include<iostream>
#define MAX_NODES 100
// the maximum number of nodes in the graph
#define INF 2147483646 // represents infity
#define UNINITIALIZED -1 // value for node with no parent
using namespace std;
// represents the capacities of the edges
int capacities[MAX_NODES][MAX_NODES];
// shows how much flow has passed through an edge
int flowPassed[MAX_NODES][MAX_NODES];
// represents the graph. The graph must contain the negative edges too!
vector<int> graph[MAX_NODES];
//shows the parents of the nodes of the path built by the BFS
int parentsList[MAX_NODES];
//shows the maximum flow to a node in the path built by the BFS
int currentPathCapacity[MAX_NODES];
23
The Max-flow algorithm - core
 The “heart” of the algorithm:
int edmondsKarp(int startNode, int endNode) {
int maxFlow=0;
while(true) {
int flow=bfs(startNode, endNode);
if(flow==0) break;
maxFlow +=flow;
int currentNode=endNode;
while(currentNode != startNode) {
int previousNode = parentsList[currentNode];
flowPassed[previousNode][currentNode] += flow;
flowPassed[currentNode][previousNode] -= flow;
currentNode=previousNode;
}
}
return maxFlow;
}
24
The Max-flow algorithm –
Breadth-first search
 Breadth-first
search
int bfs(int startNode, int endNode)
{
memset(parentsList, UNINITIALIZED, sizeof(parentsList));
memset(currentPathCapacity, 0, sizeof(currentPathCapacity));
queue<int> q;
q.push(startNode);
parentsList[startNode]=-2;
currentPathCapacity[startNode]=INF;
. . .
25
The Max-flow algorithm –
Breadth-first search
...
while(!q.empty()) {
int currentNode = q.front(); q.pop();
for(int i=0; i<graph[currentNode].size(); i++) {
int to = graph[currentNode][i];
if(parentsList[to] == UNINITIALIZED
&& capacities[currentNode][to] - flowPassed[currentNode][to] > 0) {
parentsList[to] = currentNode;
currentPathCapacity[to] = min(currentPathCapacity[currentNode],
capacities[currentNode][to] - flowPassed[currentNode][to]);
if(to == endNode) return currentPathCapacity[endNode];
q.push(to);
}
}
}
return 0;
}
26
Run-time estimation
 Breaking
down the algorithm:
 The BFS will cost O(E) operations to find a path
on each iteration
 We will have total O(VE) path augmentations
(proved with Theorem and Lemmas)
 This gives us total run-time of O(VE*E)
27
Run-time estimation

There are other algorithms that can run in O(V³)
time but are far more complicated to implement

! Note - this algorithm can also run in O(V³) time
for sparse graphs
28
The Max-flow algorithm
 Perks of using the Edmonds-Karp algorithm
 Runs relatively fast in sparse graphs
 Represents a refined version of the FordFulkerson algorithm
 Unlike the Ford-Fulkerson algorithm, this will
always terminate
 It is relatively simple to implement
29
Summary

Many problems can be transformed to a max-flow
problem. So keep your eyes open!

The Edmonds-Karp algorithm is:
 fairly fast for sparse graphs – O(V³)
 easy to implement
 runs in O(VE²) time
30
Summary

Don’t forget to add the reverse edges to your
graph!

The algorithm
 Looks for augmenting path
from source to sink on each
iteration
 Maximum flow == smallest residual capacity of an
edge in that path
31
Resources
 Video lectures (in kind of English)
 http://www.youtube.com/watch?v=J0wzih3_5Wo

http://en.wikipedia.org/wiki/Maximum_flow_problem

http://en.wikipedia.org/wiki/Edmonds%E2%80%93Kar
p_algorithm

http://en.wikipedia.org/wiki/Matching_(graph_theory)
 Nakov’s book: Programming = ++Algorithms;
32
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