network flow - utkweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › network flow.pdf ·...
TRANSCRIPT
![Page 2: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/2.jpg)
Table of Contents
Basic Definitions
Motivation and History
Theory
Max-Flow Min Cut
Applications
Open Problems
Homework
References
2
![Page 3: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/3.jpg)
Table of Contents
Basic Definitions
Motivation and History
Theory
Max-Flow Min Cut
Applications
Open Problems
Homework
References
3
![Page 4: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/4.jpg)
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
![Page 5: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/5.jpg)
Definition
5
![Page 6: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/6.jpg)
Definition
In the next 20 minutes we will learn:
How much more flow the above network can take? 6
![Page 7: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/7.jpg)
Table of Contents
Basic Definitions
Motivation and History
Theory
Max-Flow Min Cut
Applications
Open Problems
Homework
References
7
![Page 8: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/8.jpg)
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
![Page 9: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/9.jpg)
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
![Page 10: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/10.jpg)
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
![Page 11: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/11.jpg)
Frank S. Ross
• Chief of the Army’s Transportation Corps in
Europe
11
![Page 12: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/12.jpg)
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
![Page 13: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/13.jpg)
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
![Page 14: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/14.jpg)
Table of Contents
Basic Definitions
Motivation and History
Theory
Max-Flow Min Cut
Applications
Open Problems
Homework
References
14
![Page 15: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/15.jpg)
Residual Network
15
![Page 16: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/16.jpg)
Residual Network
16
![Page 17: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/17.jpg)
Augmenting Path
17
![Page 18: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/18.jpg)
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
![Page 19: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/19.jpg)
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
![Page 20: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/20.jpg)
Cut
20
![Page 21: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/21.jpg)
Table of Contents
Basic Definitions
Motivation and History
Theory
Max-Flow Min Cut
Applications
Open Problems
Homework
References
21
![Page 22: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/22.jpg)
Max-Flow Min-Cut
22
![Page 23: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/23.jpg)
Max-Flow Min-cut
23
![Page 24: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/24.jpg)
Max-Flow Min-cut
24
![Page 25: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/25.jpg)
Max-Flow Min-cut
25
![Page 26: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/26.jpg)
Max-Flow Min-cut
26
![Page 27: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/27.jpg)
Max-Flow Min-cut
27
![Page 28: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/28.jpg)
Max-Flow Min-cut
28
![Page 29: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/29.jpg)
Max-Flow Min-cut
29
![Page 30: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/30.jpg)
Max-Flow Min-cut
30
![Page 31: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/31.jpg)
Max-Flow Min-Cut Theorem
31
![Page 32: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/32.jpg)
Max-Flow Min-Cut
32
![Page 33: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/33.jpg)
Max-Flow Min-Cut Algorithm
33
![Page 34: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/34.jpg)
Max-Flow Min-Cut Algorithm 2,1
4,4 3,3 1,1
34
![Page 35: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/35.jpg)
Max-Flow Min-Cut Algorithm 2,1
4,4 3,3 1,1
35
![Page 36: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/36.jpg)
Max-Flow Min-Cut Algorithm 2,1
4,4 3,3 1,1
2,1
4,4 3,3 1,1
36
![Page 37: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/37.jpg)
Max-Flow Min-Cut Algorithm 2,1
4,4 3,3 1,1
2,1
4,4 3,3 1,1
37
![Page 38: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/38.jpg)
Max-Flow Min-Cut Algorithm 2,1
4,4 3,3 1,1
2,2
4,4 3,3 1,1
38
![Page 39: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/39.jpg)
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
![Page 40: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/40.jpg)
Max-Flow Min-Cut Algorithm
40
![Page 41: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/41.jpg)
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
![Page 42: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/42.jpg)
Table of Contents
Basic Definitions
Motivation and History
Theory
Max-Flow Min Cut
Applications
Open Problems
Homework
References
42
![Page 43: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/43.jpg)
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
![Page 44: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/44.jpg)
Maximum Bipartite Matching The solution to the maximum flow problem gives us a solution to the
maximum bipartite matching problem
44
![Page 45: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/45.jpg)
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
![Page 46: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/46.jpg)
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
![Page 47: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/47.jpg)
Circulation
47
![Page 48: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/48.jpg)
Circulation
The original circulation problem has a solution iff its network flow problem has
a maximum flow value D
48
![Page 49: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/49.jpg)
Circulation
49
![Page 50: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/50.jpg)
Table of Contents
Basic Definitions
Motivation and History
Theory
Max-Flow Min Cut
Applications
Open Problems
Homework
References
50
![Page 51: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/51.jpg)
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)
51
![Page 52: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/52.jpg)
Table of Contents
Basic Definitions
Motivation and History
Theory
Max-Flow Min Cut
Applications
Open Problems
Homework
References
52
![Page 53: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/53.jpg)
Homework
3,3
4,3 5,3 4,4
3,3
53
![Page 54: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/54.jpg)
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
![Page 55: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/55.jpg)
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
![Page 56: Network Flow - UTKweb.eecs.utk.edu › ~cphill25 › cs594_spring2016 › Network Flow.pdf · Definition Flow Network N is a directed graph where each edge has a capacity and each](https://reader033.vdocument.in/reader033/viewer/2022052802/5f1ea936d690d7743b54334a/html5/thumbnails/56.jpg)
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