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Table of Contents
Basic Definitions
Motivation and History
Theory
Max-Flow Min Cut
Applications
Open Problems
Homework
References
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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.
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Definition
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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.
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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
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Frank S. Ross
• Chief of the Army’s Transportation Corps in
Europe
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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
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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
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Table of Contents
Basic Definitions
Motivation and History
Theory
Max-Flow Min Cut
Applications
Open Problems
Homework
References
14
Residual Network
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Residual Network
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Augmenting Path
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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
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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).
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Cut
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Table of Contents
Basic Definitions
Motivation and History
Theory
Max-Flow Min Cut
Applications
Open Problems
Homework
References
21
Max-Flow Min-Cut
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Max-Flow Min-cut
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Max-Flow Min-cut
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Max-Flow Min-cut
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Max-Flow Min-cut
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Max-Flow Min-cut
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Max-Flow Min-cut
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Max-Flow Min-cut
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Max-Flow Min-cut
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Max-Flow Min-Cut Theorem
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Max-Flow Min-Cut
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Max-Flow Min-Cut Algorithm
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Max-Flow Min-Cut Algorithm 2,1
4,4 3,3 1,1
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Max-Flow Min-Cut Algorithm 2,1
4,4 3,3 1,1
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Max-Flow Min-Cut Algorithm 2,1
4,4 3,3 1,1
2,1
4,4 3,3 1,1
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Max-Flow Min-Cut Algorithm 2,1
4,4 3,3 1,1
2,1
4,4 3,3 1,1
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Max-Flow Min-Cut Algorithm 2,1
4,4 3,3 1,1
2,2
4,4 3,3 1,1
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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
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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.
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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
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Maximum Bipartite Matching The solution to the maximum flow problem gives us a solution to the
maximum bipartite matching problem
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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.
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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.
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Circulation
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Circulation
The original circulation problem has a solution iff its network flow problem has
a maximum flow value D
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Circulation
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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 4-
flow. Tutte(1966)
• Every 2-edge- connected graph without 3-edge cuts has a 3-flow.
Tutte(1972)
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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 4,4
3,3
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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.
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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
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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
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