Network Flow
Helia Zandi
hzandi@vols.utk.edu
4/20/2016
1
Table of Contents
 Basic Definitions
 Motivation and History
 Theory
 Max-Flow Min Cut
 Applications
 Open Problems
 Homework
 References
2
Table of Contents
 Basic Definitions
 Motivation and History
 Theory
 Max-Flow Min Cut
 Applications
 Open Problems
 Homework
 References
3
Definition
Flow Network N is a directed graph where each edge has a capacity and
each edge receives a flow. The amount of flow on an edge cannot exceed
the capacity of the edge.
In a network, the vertices are called nodes and the edges are called arcs.
The capacity function c of network N is a nonnegative function on E(D).
If a = (u, v) is an arc of D, then c(a) = c(u, v) is called the capacity of a.
The diagraph D is called the underlying diagraph of N.
There are two special vertices in a network s and t, called source and
sink, respectively. Source has only outgoing flow and sink has only
incoming flow.
4
Definition
5
Definition
In the next 20 minutes we will learn:
How much more flow the above network can take?
6
Table of Contents
 Basic Definitions
 Motivation and History
 Theory
 Max-Flow Min Cut
 Applications
 Open Problems
 Homework
 References
7
Motivation
 The original inspiration comes from
the Cold War.
 During the Cold War, the US
military was interested in knowing
what was the minimum number of
places on the railroad system they
could bomb that would completely
and accurately prevent movement
between the Soviet Union and
Eastern Europe.
8
Motivation
 The US Air Force requested a
secret report which was written in
1955 by T.E. Harris and F.S. Ross
entitled Fundamentals of a Method
for Evaluating Rail Net Capacities.
 The max flow problem was
formally defined in this report.
 Harris and Ross did not find a
method that was guaranteed to find
an optimal solution. The technique
they described is to use a greedy
algorithm of pushing as much flow
as possible through the network
until there is a bottleneck( a vertex
that has more flowing coming in to
it than is able to leave)
9
Theodore Edward "Ted" Harris
• 11 January 1919 – 3 November 2005
• An American Mathematician known for his
research in stochastic process
• Mathematics department head at RAND
corporation
• Professor at University of Southern California
• Harris inequality in statistic physic and
percolation theory is named after him
• In 1989 he received an honorary doctorate from
Chalmers Institute of Technology, Sweden
10
Frank S. Ross
• Chief of the Army’s Transportation Corps in
Europe
11
Lester Randolph Ford, Jr
• Born September 23, 1927, Huston
• An American Mathematician specializing
in network flow problems
• Developed Ford-Fulkerson algorithm for solving
max flow problem
• Ford also developed the Bellman-Ford
algorithm for finding shortest path in graphs
12
Delbert Ray Fulkerson
• August 14, 1924 – January 10, 1976
• An American Mathematician
• Co-developed Ford-Fulkerson algorithm
• In 1979, Fulkerson prize was established which
is now awarded every three years for outstanding
papers in discrete mathematics
• Remained at Cornell until he committed suicide
13
Table of Contents
 Basic Definitions
 Motivation and History
 Theory
 Max-Flow Min Cut
 Applications
 Open Problems
 Homework
 References
14
Residual Network
15
Residual Network
16
Augmenting Path
17
Augmenting Path
In other words it is a path which can admit additional flow from
the source to the sink (all edges along the path have residual capacity)
Augmenting Path
We will prove that if there is NO
augmenting path in a network
The flow is maximum
18
Cut
A Cut is a partition of V into sets S and T such that s ∈ S and T = V - S has t ∈ T.
The net flow across the cut is f(S,T) and the capacity of the cut is c(S,T).
19
Cut
20
Table of Contents
 Basic Definitions
 Motivation and History
 Theory
 Max-Flow Min Cut
 Applications
 Open Problems
 Homework
 References
21
Max-Flow Min-Cut
22
Max-Flow Min-cut
23
Max-Flow Min-cut
24
Max-Flow Min-cut
25
Max-Flow Min-cut
26
Max-Flow Min-cut
27
Max-Flow Min-cut
28
Max-Flow Min-cut
29
Max-Flow Min-cut
30
Max-Flow Min-Cut Theorem
31
Max-Flow Min-Cut
32
Max-Flow Min-Cut Algorithm
33
Max-Flow Min-Cut Algorithm
2,1
4,4
3,3
1,1
34
Max-Flow Min-Cut Algorithm
2,1
4,4
3,3
1,1
35
Max-Flow Min-Cut Algorithm
2,1
4,4
3,3
1,1
2,1
4,4
3,3
1,1
36
Max-Flow Min-Cut Algorithm
2,1
4,4
3,3
1,1
2,1
4,4
3,3
1,1
37
Max-Flow Min-Cut Algorithm
2,1
4,4
3,3
1,1
2,2
4,4
3,3
1,1
38
Max-Flow Min-Cut Algorithm
2,1
4,4
3,3
1,1
2,2
4,4
3,3
1,1
Max flow value = 6
39
Max-Flow Min-Cut Algorithm
40
Multiple sources and Sinks
Problems with multiple sources and sinks can be reduced to the
single source/sink case.
In these cases a supersource is introduced. This consists of a vertex
connected to each of the sources with edges of infinite capacity. The
same definition applies to supersink.
41
Table of Contents
 Basic Definitions
 Motivation and History
 Theory
 Max-Flow Min Cut
 Applications
 Open Problems
 Homework
 References
42
Applications
 Modelled after transportation in a network
 The power is in efficient solutions to problems:
 bipartite matching
 edge-disjoint paths
 vertex-disjoint paths
 Communication network
 scheduling
 circulation
 image segmentation
 weighted bipartite matching
 several “easier” proofs in graph theory
 Theorem of Hall
 Theorem of Menger
43
Maximum Bipartite Matching
The solution to the maximum flow problem gives us a solution to the
maximum bipartite matching problem
44
Edge-disjoint Paths
Given a digraph G=(V,E) and two nodes s and t , what is the maximum
number of edge-disjoint s-t paths.
The maximum flow is equal to the maximum number of edge-disjoint paths.
45
Graph Connectivity
Given a digraph G=(V,E) and two nodes and t , what is the minimum
number of edges whose removal disconnects s and t
Menger’s Theorem. The maximum number of edge-disjoint s-t
paths is equal to the minimum number of edges that disconnects s
from t
Theorem. A graph is n-edge-connected, if and only if every two
vertices of G is connected by at least n edge-disjoint path.
46
Circulation
47
Circulation
The original circulation problem has a solution iff its network flow problem has
a maximum flow value D
48
Circulation
49
Table of Contents
 Basic Definitions
 Motivation and History
 Theory
 Max-Flow Min Cut
 Applications
 Open Problems
 Homework
 References
50
Open Problems
• Every 2-edge connected graph has a 5-flow. Tutte(1954)
• Every 2-edge- connected graph with no Peterson minor has a 4flow. Tutte(1966)
• Every 2-edge- connected graph without 3-edge cuts has a 3-flow.
Tutte(1972)
51
Table of Contents
 Basic Definitions
 Motivation and History
 Theory
 Max-Flow Min Cut
 Applications
 Open Problems
 Homework
 References
52
Homework
3,3
4,3
5,3
3,3
4,4
53
Homework
3. Let N be a network and f a flow in N. Prove that the value of flow in N
equals the net value into the sink t of network N.
54
References
•
•
•
•
•
•
•
•
•
•
•
•
•
Applied and Algorithmic Graph Theory, Gary Chartland, Orturd R. Oellermann, 1993
Graph Theory, J.A. Bondy, U.S.R. Murty, 2008
https://en.wikipedia.org/wiki/Flow_network
http://www.me.utexas.edu/~jensen/network_02/announce.html
http://blogs.cornell.edu/info4220/2015/03/10/the-origin-of-the-study-of-network-flow/
https://courses.cs.washington.edu/courses/csep521/13wi/video/archive/html5/video.ht
ml?id=csep521_13wi_9
http://faculty.ycp.edu/~dbabcock/PastCourses/cs360/lectures/lecture24.html
http://slideplayer.com/slide/4705331/
http://homepages.cwi.nl/~lex/files/histtrpclean.pdf
http://courses.cs.vt.edu/~cs4104/murali/Fall09/lectures/lecture-20-network-flow.pdf
http://www.cs.princeton.edu/courses/archive/spr04/cos226/lectures/maxflow.4up.pdf
http://staff.ustc.edu.cn/~csli/graduate/algorithms/book6/chap27.htm
http://www.me.utexas.edu/~jensen/network_02/topic_pages/sbayti/history.html#Genera
lized Network Applications
55
References
•
•
•
•
http://www.es.ele.tue.nl/education/5MC10/9flow.pdf
http://www.geeksforgeeks.org/find-edge-disjoint-paths-two-vertices/
https://en.wikipedia.org/wiki/Circulation_problem
http://www2.hawaii.edu/~nodari/teaching/s15/Notes/Topic-20.html
56