CHAPTER 2
Graph Foundations of Network
Modelling
M. Pickavet and C. Develder
1
Outline
0. Introduction
1. Definition and notations
1.1 Graph
1.2 Digraph
2. Minimum spanning tree problem
3. Shortest path problem
3.1 in unweighted (di)graphs
3.2 in weighted (di)graphs (positive weights)
3.3 in weighted (di)graphs (general weights)
3.4 all-pairs shortest path
Graph foundations of
network modelling 2
Outline
4. Maximum flow problem
4.1 Network flow
4.2 Problem definition
4.3 Residual network concept
4.4 Flow augmenting path algorithm
4.5 Max-flow min-cut theorem
4.6 Application: calculation of connectivity
5. Minimum cost flow problem
5.1 Introduction
5.2 Problem definition
5.3 Residual network
5.4 Busacker-Gowen algorithm
5.5 Several supply/demand vertices
6. Extension: multi-commodity flow problems
Graph foundations of
network modelling 3
Outline
0.Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
4. Maximum flow problem
5. Minimum cost flow problem
6. Extension: multi-commodity flow
problems
Graph foundations of
network modelling 4
Why graph foundations in a MM network course?
efficient transfer of information across
a communication network
network flow problem
flow algorithm
Examples :
• shortest path problem
• maximum flow problem
• minimum cost flow problem
Graph foundations of
network modelling 5
Abstract modelling of communication networks
What must be modeled ?
all the relevant data of the real network:
• topology
• transfer of information
• different parameters
• cost of equipment
• capacity of equipment
•…
Graph foundations of
network modelling 6
Outline
0. Introduction
1. Definition and notations
2.
3.
4.
5.
6.
1.1 Graph
1.2 Digraph
Minimum spanning tree problem
Shortest path problem
Maximum flow problem
Minimum cost flow problem
Extension: multi-commodity flow problems
Graph foundations of
network modelling 7
Terminology and notations
NODE
v1
e3
v4
IP router,
SDH cross-connect,
LINK
vATM
switch,
2
telephoneSTM-N,
switch,...
optical fiber, ...
e1
e2
e4
e5
v3
• incidence: I (v1) = {e1,e2,e3}
• vertex set V
• edge set E
G =(V, E)
• graph order p = #V = 4
• graph size q = #E = 5
• G is a (p, q) - graph
• e = uv
• adjacency: A(v2) = {v1,v3}
• degree: d(v3) = 3
Subgraph G[W] : subset W of vertices of G + edges incident to vertices of subgraph W only
example : {v1, v2, v3}
==> {e1, e2, e4}
Graph foundations of
network modelling 8
Paths and Cycles
walk w = (v0, v1, …, vn-1, vn) such that for i = 0, 1, …, n-1, vivi+1  E
v1
e3
v4
e1
v2
e2
e4
e5
v3
Special types of walks : paths / cycles
(v1, v4, v3) is a path with length 2
(v1, v4, v3, v1) is a cycle with length 3
Walk with all vertices distinct
Walk with all vertices but
the first and the last distinct
Graph foundations of
network modelling 10
Cut and Cutset
v5
e8
v1
e3
v4
e1
e6
v2
e7
e2
e4
e5
v3
G is connected : " vertex pairs {u,
v}, a path exists between u and v
• cut, e.g. {e2, e3, e5}
• cutset (= minimal cut), e.g. {e3, e5}
• vertex-cut, e.g. {v1, v2, v3}
• vertex-cutset (= minimal vertex-cut),
e.g. {v1, v3}
Graph foundations of
network modelling 11
Connectivity
v5
e8
v1
e3
v4
e1
e6
v2
e6
e7
e2
e4
e5
v5
v3
v1
e3
v4
A graph G
with edge-connectivity l(G) = 2
(= minimum cardinality cutset = # {e3,e5})
and vertex-connectivity k(G) = 2
(= minimum cardinality vertex-cutset = # {v1,v3})
e1
v2
e2
e4
e5
v3
A graph G with bridge e6
(note that v5 is not a cutvertex)
l(G) = 1, k(G) = 1
Graph foundations of
network modelling 12
Connectivity
EXERCISE
v6
v5
v8
v7
v1
v3
v4
edge-connectivity l(G) = ?
vertex-connectivity k(G) = ?
v2
Graph foundations of
network modelling 13
Outline
0. Introduction
1. Definition and notations
2.
3.
4.
5.
6.
1.1 Graph
1.2 Digraph
Minimum spanning tree problem
Shortest path problem
Maximum flow problem
Minimum cost flow problem
Extension: multi-commodity flow problems
Graph foundations of
network modelling 14
Digraph
Directed Graph
v1
a3
v4
a1
• vertex set V
• arc set A
v2
D = (V,A)
a2
a4
a5
• digraph order = #V = 4
• digraph size = #A = 5
v3
• from - incidence : I (v1) = {a1, a3}
• to - incidence : I’(v4) = {a3, a5}
• from - adjacency : A(v3) = {v1,v4}
• to - adjacency : A’(v2) = {v1}
• out - degree : d(v3) = 2
• in - degree : d’(v1) = 1
Graph foundations of
network modelling 15
Paths and Cycles
walk w = (v0, v1, …, vn-1, vn) such that for i = 0, 1, …, n-1, (vi,vi+1)  A
semiwalk = (v0, v1, …, vn-1, vn) such that for i = 0, 1, …, n-1, (vi,vi+1)  A  (vi+1,vi A
v1
a3
v4
a1
v2
a2
a4
a5
v3
(v1, v2, v3) is a path with length 2
(v1, v2, v3, v1) is a cycle with length 3
(v1, v4, v3) is a semipath with length 2
(v1, v4, v3, v1) is a semicycle with length 3
Graph foundations of
network modelling 17
Strong and weak connectedness
v5
v5
v1
v2
v1
v2
v4
v3
v4
v3
this digraph is strongly connected
(u,v)- AND (v,u)-paths
this digraph is weakly connected
(u,v)- AND/OR (v,u)-path
there are no paths terminating in v3
Graph foundations of
network modelling 18
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
4. Maximum flow problem
5. Minimum cost flow problem
6. Extension: multi-commodity flow problems
Graph foundations of
network modelling 19
Tree and spanning tree
v5
v5
v1
v4
v2
v3
a tree T=(V,F)
• T is connected
• T does not contain cycles
 T contains #V-1 edges
v1
v2
v4
v3
a spanning tree T=(V,F) of G
• T is a tree
• T has the same vertex set as G
Graph foundations of
network modelling 20
MST: algorithm of Kruskal
weighted graph
c(e) = weight of edge e
weight of weighted graph = sum of weights of all edges
minimum spanning tree = spanning tree with minimum weight (in a weighted graph)
Mackinaw City
Traverse City
3
105
4
170
306
85
245
135
5
180
6
Benton Harbor
{insertion of edge 56 with weight 85 into F}
P = { {1}, {2}, {3}, {4}, {5,6} }
206
210
140
Grand
Rapids
P = { {1}, {2}, {3}, {4}, {5}, {6} }
2
270
137
139
Sagninaw
{insertion of edge 34 with weight 105 into F}
P = { {1}, {2}, {3,4}, {5,6} }
{insertion of edge 12 with weight 112 into F}
P = { {1,2}, {3,4}, {5,6} }
112
300
1
Detroit
{insertion of edge 25 with weight 135 into F}
P = { {1,2,5,6}, {3,4} }
{insertion of edge 45 with weight 140 into F}
Graph foundations of
P = { {1,2,3,4,5,6} }
network modelling 21
Minimum spanning tree: applications
Characteristics MST:
• network infrastructure connecting all ‘entities’
• minimal cost  price is key
• no fault-tolerance  not important ?
• unique path  unequivocal routing, cycles avoided
Applications area:
• communication networks:
• minimal topology
• virtual tree network
• other networking situations:
• electricity network
• sewer system network
•…
Graph foundations of
network modelling 22
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
3.1
3.2
3.3
3.4
In unweighted (di)graphs
In weighted (di)graphs (positive weights)
In weighted (di)graphs (general weights)
All-pairs shortest path problem
4. Maximum flow problem
5. Minimum cost flow problem
6. Extension: multi-commodity flow problems
Graph foundations of
network modelling 24
Distance in unweighted graph
The length of a path in an unweighted graph is
equal to the number of edges in the path (=‘hopcount’).
The distance d(u,v) between two vertices u,v in a graph G = (V,E)
is equal to the length of a shortest u,v-path in G,
if such a path exists (if not: d(u,v) = ).
Graph foundations of
network modelling 25
Example : the Moore Algorithm
s
v1
G:
s
v2
v4
v3
v1
Labels
0
v5
v2
1
v3
2
v6
v4
v5
v7
v8
v9
v6
v10
3
v7
v10
v8

v9
s
0
v1

v2

v3

v4

v5

v6

v7

v8

v9 v10

0
1
1


1




0
1
1
2
2
1
2


2
0
1
1
2
2
1
2
3

2
Graph foundations of
network modelling 26
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
3.1
3.2
3.3
3.4
In unweighted (di)graphs
In weighted (di)graphs (positive weights)
In weighted (di)graphs (general weights)
All-pairs shortest path problem
4. Maximum flow problem
5. Minimum cost flow problem
6. Extension: multi-commodity flow problems
Graph foundations of
network modelling 27
Distance in weighted graph
Weighted Graph : c : E  R : map every edge to a real number
Length of a path : sum of length of its edges
Distance between two vertices s,t : length of shortest s,t path
Graph foundations of
network modelling 28
Algorithm of Dijkstra
Assumption :
all weights positive !!
µ0=s
8
13
v6
v2
9
11
7
v7
v4
17
14
shortest path tree
v5
16
temporal branches
new branches
5
v3
l(µ0)
v2
v3
v4
v5
v6
v7
add to P
0
(,-)
(,-)
(,-)
(,-)
(,-)
(,-)
µ0
(13,µ0)
(,-)
(16,µ0)
(8,µ0)
(,-)
(,-)
v5
(13,µ0)
(25,v5)
(15,v5)
(,-)
(,-)
v2
(25,v5)
(15,v5)
(,-)
(,-)
v4
(,-)
(,-)
v3
(,-)
(,-)
(20,v4)
Graph foundations of
network modelling 29
Dijkstra
EXERCISE
v1
2
1
1
u0
3
v4
v6
v2
4
2
5
6
3
v5
4
5
1
2
d(u0,v5) ?
1. op zicht
2. m.b.v. Dijkstra
v3
2
v7
3
v8
Graph foundations of
network modelling 31
Dijkstra
EXERCISE
Graph foundations of
network modelling 32
Algorithm of Dijkstra: application
Router C :
Link-State Database
[AB,AE]
incoming
link state
packets
[AB,BD,BC]
[BD,CD,DE]
[AE,DE]
Link
AB
AE
BD
BC
CD
DE
Cost
1
1
1
1
1
1
knowledge of
network topology
From
A
A
B
B
D
D
Dijkstra :
shortest paths
A
E
B
LSA E {E-D, E-A}
LSA E {E-D, E-A}
LSA E {E-D, E-A}LSA E {E-D, E-A}
C
Router C :
Routing Table
Dest.
D
LSA E {E-D, E-A}
A
B
D
E
Next
hop
B
Direct
Direct
D
Interface
BC
BC
CD
CD
Graph foundations of
network modelling 33
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
3.1
3.2
3.3
3.4
In unweighted (di)graphs
In weighted (di)graphs (positive weights)
In weighted (di)graphs (general weights)
All-pairs shortest path problem
4. Maximum flow problem
5. Minimum cost flow problem
6. Extension: multi-commodity flow problems
Graph foundations of
network modelling 34
The algorithm of Ford, Bellman and Moore
Result with Dijkstra:
l(v1) = 3
4
l(v2) = 4
v2
1
3
s
l(s) = 0
- 4
v1
1
2
1
v4
l(v4) = 1
1
l(v1)  l(v2) + c(v2v1) is violated
 l(v1) is updated and set to l(v1) = l(v2) + c(v2v1)
=0
v3
t
l(t) = 5
4
l(v3) = 2
Principle : label correcting algorithm
Let l(i), iV be a set of labels. These labels represent the shortest path distances
from the source vertex s if they satisfy the following conditions:
(i)
l(s) = 0;
(ii)
l(i) is the length of some (s,i)-path;
(iii)
l(j)  l(i) + c(ij), "(i,j) A
Graph foundations of
network modelling 35
The algorithm of Ford, Bellman and Moore
(1)
l(s) := 0 ; "vV\{s} do l(v) :=  end;
(2)
insert s into List;
(3)
while List is not empty do
l(s)=0, l(v1)=l(v2)=…=l(t)=
List={s}
(4)
select the first element v from List and remove it from List;
(5)
"wA(v) do
(6)
A(v)={v1,v4}
A(v)={v4}
if l(w) > l(v) + c(vw)
l(v1)=4,
l(v4)=1 l(v4)=1
(7)
l(w) := l(v) + c(vw);
(8)
List;
if w is not yet an element of List, insert w at the back of
(9)
end;
(10) end;
A(v)=from-adjacency
in digraph
v=s, List
} }
v=v1,
List=={ {v4
4
l(s)=0
- 4
v1
v2
1
3
s
List = {v1,v4 }
List = {v4 }
2
1
Extension : negative cycles ?
1
t
4
v4
v3
1
Graph foundations of
network modelling 36
Ford-Bellman-Moore
EXERCISE
v1
2
-1
v4
3
2
v3
-6
5
1
u0
v2
4
6
v5
4
3
5
1
v6
2
v7
3
v8
Graph foundations of
network modelling 38
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
3.1
3.2
3.3
3.4
In unweighted (di)graphs
In weighted (di)graphs (positive weights)
In weighted (di)graphs (general weights)
All-pairs shortest path problem
4. Maximum flow problem
5. Minimum cost flow problem
6. Extension: multi-commodity flow problems
Graph foundations of
network modelling 39
All-pairs shortest path problem
Until now:
• Shortest path from s to t
• Side effect: shortest path tree from s to all other nodes
Current subsection:
Shortest path between ALL PAIRS OF NODES
v1
Possible approaches:
8
• p times Dijkstra
13
v5
16
• Dedicated algorithm: Floyd
v2
7
11
v4
17
14
5
v3
Graph foundations of
network modelling 40
Algorithm of Floyd
Step 0: direct paths distances
Assumption :
all weights positive !!
Step 1: x – v1 – y better ?
v1
v2
v3
v4
v5
v1
0
13

16
8
11
v2
13
0
14
29
11
5
17
v3

14
0
5
17
5
0
7
v4
16
29
5
0
7
17
7
0
v5
8
11
17
7
0
v1
v2
v3
v4
v5
v1
0
13

16
8
v2
13
0
14
?
v3

14
0
v4
16

v5
8
11
v1
Step 2: x – v2 – y better ?
v1
v2
v3
v4
v1
0
13
27
16
8
v2
13
0
14
29
11
v3
27
14
0
5
17
v4
16
29
5
0
7
v5
8
11
17
7
0
8
v5
13
v2
v5
16
7
11
v4
17
14
5
v3
Graph foundations of
network modelling 41
Algorithm of Floyd
Step 3: x – v3 – y better ?
v1
v2
v3
v4
v1
0
13
27
16
8
v2
13
0
14
19
v3
27
14
0
v4
16
19
v5
8
11
Step 4: x – v4 – y better ?
v5
v1
v2
v3
v4
v5
v1
0
13
21
16
8
11
v2
13
0
14
19
11
5
17
v3
21
14
0
5
12
5
0
7
v4
16
19
5
0
7
17
7
0
v5
8
11
12
7
0
v1
Step 5: x – v5 – y better ?
v1
v2
v3
v4
v1
0
13
20
15
8
v2
13
0
14
18
11
v3
20
14
0
5
12
v4
15
18
5
0
7
v5
8
11
12
7
0
8
v5
13
v2
v5
16
7
11
v4
17
14
5
v3
Graph foundations of
network modelling 42
Algorithm of Floyd
(1)
D(0) = A;
(2) for step := 1 to p do
(3)
(4)
for vi := v1 to vp do
for vj := v1 to vp do
(5)
(6)
(7)
di,j(step) := min {di,j(step-1) , di,vstep(step-1) + dvstep,j(step-1) }
end;
end;
(8) end.
v1
8
Implementation effort: small
13
v5
16
Time complexity: O(p3)
 p times Dijkstra: p x O(p2)
v2
7
11
v4
17
14
5
v3Graph foundations of
network modelling 43
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
4. Maximum flow problem
4.1
4.2
4.3
4.4
4.5
4.6
Network flow
Problem definition
Residual network concept
Flow augmenting path algorithm
Max-flow min-cut theorem
Application: calculation of connectivity
5. Minimum cost flow problem
6. Extension: multi-commodity flow problems
Graph foundations of
network modelling 44
Network flow models
abstract modelling of a communication network
 transfer of information  flow
 topology  directed graph (digraph) D = (V,A)
 every arc a  A has associated numerical values
 cost c(a)  weighted digraph
 capacity u(a)  capacitated digraph
network N = weighted, capacitated digraph (V,A)
Graph foundations of
network modelling 45
Network flow-associated constraints
flow  function f (.) : A  Z+ or R+
subject to the following constraints:
flow
conservation
I’(v)
v
capacity
constraints
I(v)
• supply vertex : b(v) > 0 (source)
• demand vertex : b(v) < 0 (sink)
u(a) = capacity of link
Graph foundations of
network modelling 46
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
4. Maximum flow problem
4.1
4.2
4.3
4.4
4.5
4.6
Network flow
Problem definition
Residual network concept
Flow augmenting path algorithm
Max-flow min-cut theorem
Application: calculation of connectivity
5. Minimum cost flow problem
6. Extension: multi-commodity flow problems
Graph foundations of
network modelling 47
Problem definition : maximum flow
2
v1
5
1
Source s
3
v4
v2
1
3
4
1
v3
t Sink
3
i
capacity
j
u(ij)
Graph foundations of
network modelling 48
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
4. Maximum flow problem
4.1
4.2
4.3
4.4
4.5
4.6
Network flow
Problem definition
Residual network concept
Flow augmenting path algorithm
Max-flow min-cut theorem
Application: calculation of connectivity
5. Minimum cost flow problem
6. Extension: multi-commodity flow problems
Graph foundations of
network modelling 49
Residual network concept
flow
i
residual capacity
capacity
[ f °(ij),u(ij)]
j
r(ij)
i
3
3
0,4]
[3,4
Source
0,5]
[5,5
3,5
[2,3
0,3]
1
[2,2
0,2]
flow=3
flow=2
j
4
[0,1]
2
original network
1
Sink
5
3
2 1
1
4
1
2
2
residual network
Graph foundations of
network modelling 50
Residual Network
EXERCISE
i
[ f °(ij),u(ij)]
j
[4,4]
[1,2]
[3,5]
[0,1]
[1,1]
[1,3]
[1,2]
Graph foundations of
network modelling 51
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
4. Maximum flow problem
4.1
4.2
4.3
4.4
4.5
4.6
Network flow
Problem definition
Residual network concept
Flow augmenting path algorithm
Max-flow min-cut theorem
Application: calculation of connectivity
5. Minimum cost flow problem
6. Extension: multi-commodity flow problems
Graph foundations of
network modelling 52
Flow Augmenting Path Algorithm
determine the maximum flow
between two vertices s and t in a network N.
3
residual network N(f) :
2
source
2
11
2 1
1
augmenting path ?
4
4
sink
d = residual capacity
of augmenting path
1
2
2
(1)
(2)
(3)
(4)
(5)
(6)
(7)
f := 0;
while ( N (f ) contains at least one augmenting path from s to t ) do
identify one augmenting path P from vertex s to vertex t;
d := min{r(ij) : ij  P};
augment the flow f with d units along P;
update N (f );
od
Graph foundations of
network modelling 53
Flow Augmenting Path Algorithm
r(ij)
i
j
3
3
5
4
4
1
4
1
4
3
Source
2
Sink
1
2
1
2
(a)
(b)
3
3
4
5
2
1
1
2
4
4
3
4
1
1
5
2
1
1
1
2
(c)
4
1
1
2
2
(d)
Graph foundations of
network modelling 54
Do we need
a
residual
network
??
[ f ij u ij ]
( ), ( )
i
Orig.
?
2
[0,2]
j
[0,2]
4
[2,2]
[2,2]
Source 1
[2,2]
3
Res.
[0,2]
i
5
2
[f (ij), u(ij)]
2
2
2
(c)
2
[2,2]
2
5
i
Orig.
4
2
3
3
r(ij)
2
2
2
(b)
j
6
3
2
j
[2,2]
4
[0,2]
1
[2,2]
6
2
(a)
2
1
4
2
1
[0,2]
j
2
2
2
6 Sink
5
r(ij)
i
Res.
[2,2]
6
[2,2]
5
[2,2]
(d)
Graph foundations of
network modelling 57
Maximum Flow Algorithm
EXERCISE
A
2
7
5
1
2
5
1
5
2
2
i
5
j
2
2
2
i
cap
5
1
3
B
capacity
j
u(ij)
Graph foundations of
network modelling 58
Maximum Flow Algorithm: applications
Characteristics max. flow solution:
• from one source to one destination
• maximal use of network BW
 bifurcation of flow
 compliant with application ? (different delays, reordering, …)
• cost of paths not taken into account
Some applications:
• maximum BW from server  client
• indication on available BW source  destination
• restoration mechanism: available spare capacity ?
Graph foundations of
network modelling 59
Maximum Flow Algorithm: applications
via spare capacity
in network
Graph foundations of
network modelling 60
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
4. Maximum flow problem
4.1
4.2
4.3
4.4
4.5
4.6
Network flow
Problem definition
Residual network concept
Flow augmenting path algorithm
Max-flow min-cut theorem
Application: calculation of connectivity
5. Minimum cost flow problem
6. Extension: multi-commodity flow problems
Graph foundations of
network modelling 61
The Maximum-Flow Minimum-Cut Theorem
correspondence between flows and cuts in a network
minimum cut problem :
“From among all cuts in the network
that separate the source and the sink,
find the cut with minimum capacity”.
Network example:
u(ij)
i
4
2
v1
1
Source s
2
j
v2
1
4
1
t Sink
3
v4
3
st : 6
v3
st : 8
Graph foundations of
network modelling 62
The Maximum-Flow Minimum-Cut Theorem
Theorem:
The maximum value of a flow between s and t
=
capacity of a minimum s,t-cut
Example: maximum s,t-flow F = 6
f(ij)
i
v1
4
Source
2
1
s
2
v4
j
v2
1
3
minimum-cut
3
1
v3
t
Sink
3
Graph foundations of
network modelling 64
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
4. Maximum flow problem
4.1
4.2
4.3
4.4
4.5
4.6
Network flow
Problem definition
Residual network concept
Flow augmenting path algorithm
Max-flow min-cut theorem
Application: calculation of connectivity
5. Minimum cost flow problem
6. Extension: multi-commodity flow problems
Graph foundations of
network modelling 65
Calculation of edge-connectivity
l(v,w) = minimum cardinality of a v,w-edge-cutset
l(G) = edge-connectivity of graph G = min{l(v,w) | v,w V }.
How to calculate l(v,w) ?
Graph G
Max flow min cut theorem :
l(v,w) = maximum flow F(v,w)
(if all link capacities = 1)
x
Residual
Network :
v
w
1 1
1
1
v
1
1
y
l(v,w) = 3
l(x,y) = 2
...
1
1
 l(G) = 2
1
1
1
1
1
1 1
w
1
l(v,w) = max. # edge-disjoint
paths between v and w
Graph foundations of
network modelling 66
Calculation of edge-connectivity
1 1
1
1
1
v
1
1
1
1
1
1
1
1
2
1 1
w
2
2
1
v
1
1
1
1
1
1
1
2
v
2
2
2
2
2
2
w
1
2
v
1
2
2
2
2
2
1
1
2
w
1
3 edge-disjoint paths
 l(v,w)=3
w
2
Graph foundations of
network modelling 67
Calculation of vertex-connectivity
k(v,w) = minimum cardinality of a v,w-vertex-cutset
k(G)=min{k(v,w) | v,w V, vw E}
How to calculate k(v,w)
Graph G
k(v,w) = max. # vertex-disjoint paths between v and w
x
• Split vertices : v’, v”
• Interconnect vertices : v’v”, v’w”, v”w’ (all link capacities = 1)
x’
v
(v,w non adjacent)
w
x”
l
1
1
1
y
k(v,w) = 2
k(G) = 2
1
l
v’
1
v”= s
1
1
1
y’
• k(v,w) = maximum flow F(v”,w’)
l
t = w’
l
w”
1
y”
Graph foundations of
network modelling 68
Calculation of vertex-connectivity
x’
x”
l
1
1
1
1
l
v’
1
v”
1
1
1
y’
w’
l
l
w”
1
y”
k(v,w) = 2
Graph foundations of
network modelling 69
Connectivity: applications
Network survivability:
•
•
•
•
to cope with single link failures:
edge-connectivity  2
to cope with single node failures:
vertex-connectivity  2
to cope with two links failing simultaneously: edge-connectivity  3
…
Graph foundations of
network modelling 70
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
4. Maximum flow problem
5. Minimum cost flow problem
5.1
5.2
5.3
5.4
5.5
Introduction
Problem definition
Residual network
Busacker-Gowen algorithm
Several supply/demand vertices
6. Extension: multi-commodity flow
problems
Graph foundations of
network modelling 71
Introduction
Example:
i
[3,5]
Server  Source
[c (ij),u (ij)]
[5,2]
v1
[1,1]
s
[2,3]
[1,1]
j
v2
[1,4]
[1,1]
t
Sink  Client
[3,3]
v4
[2,3]
v3
demand (sourcesink) = 6
Minimize the total cost of flow
with respect to how to route the flow
subject to
• the flow value on each arc respects the arc capacity
• the total flow fulfills the demand between s and t
Graph foundations of
network modelling 72
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
4. Maximum flow problem
5. Minimum cost flow problem
5.1
5.2
5.3
5.4
5.5
Introduction
Problem definition
Residual network
Busacker-Gowen algorithm
Several supply/demand vertices
6. Extension: multi-commodity flow
problems
Graph foundations of
network modelling 73
Problem definition
Optimisation model for minimum cost flow problem
Graph foundations of
network modelling 74
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
4. Maximum flow problem
5. Minimum cost flow problem
5.1
5.2
5.3
5.4
5.5
Introduction
Problem definition
Residual network
Busacker-Gowen algorithm
Several supply/demand vertices
6. Extension: multi-commodity flow
problems
Graph foundations of
network modelling 75
Residual network concept
original network N
with a certain flow f °(ij)
corresponding
residual network N( f °)
every arc (i,j) in original network  two arcs (i,j) and (j,i) in residual network
[c(ij), r(ij) = u(ij) - f °(ij)]
i
c(ij),u(ij),f(ij)
j
i
j
[-c(ij), r(ji) = f °(ij)]
Graph foundations of
network modelling 76
Residual network : example
original network N
with a certain flow f °
i
[c(ij),u(ij)]
f °(ij)
corresponding
residual network N( f °)
j
[c(ij),r(ij)]
i
3
3
[4,4]
Source 1
[2,2]
2
[1,3] 2
2
(a)
[1,2]
[4,4]
[1,4]
[-1,2]
4
[3,2]
2
j
Sink
1
[-1,2] [1,1]
4
[3,2]
[-2,2]
2
(b) Graph foundations of
network modelling 77
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
4. Maximum flow problem
5. Minimum cost flow problem
5.1
5.2
5.3
5.4
5.5
Introduction
Problem definition
Residual network
Busacker-Gowen algorithm
Several supply/demand vertices
6. Extension: multi-commodity flow
problems
Graph foundations of
network modelling 78
The algorithm of Busacker and Gowen
algorithm to determine a minimum cost flow f in a network
N from a source vertex s to a sink vertex t with value d
(demand).
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
f := 0; // initially there is no flow in the network
while ( F < d ) do
identify the shortest augmenting path P from s to t in N( f );
d:= min { min{ r(ij) : ij  P }, (d - F ) } ;
augment the flow f along P with d;
update N ( f );
F :=F + d;
od
Remarks : (i) It is assumed that a feasible flow from s to t with value d is possible.
(ii) Flow value = F =
Graph foundations of
network modelling 79
Busacker-Gowen : example
i
[c(ij),u(ij)]
(a)
[4,2]
s
[3,4]
[1,2]
(b)
v1
[4,2]
s
v4
v1
[3,4]
[-1,2]
v4
demand ( s  t ) 
j
[1,2]
v2
[1,4]
[1,2]
[-1,2]
[1,2]
[3,4]
v3
v2
[-1,2]
[1,2]
[-1,2]
f=2
(c)
cost path = 5
s
t
f=2
cost path = 7
[-1,2]
t
[4,3]
[-1,2]
v1
[-4,2]
[1,4]
[3,4]
v2
v4
(d)
[4,2]
s
[-1,2]
v1
[3,4]
[1,2]
v4
[-1,2]
[3,4]
[-1,2]
[4,3]
[3,4]
v3
d=4
[1,2]
[1,4]
[-4,2]t
[4,1]
v3
v2
[1,2]
[3,4]
v3
t
[4,3]
[1,2]
Graph foundations of
network modelling 80
Busacker-Gowen
EXERCISE
[2,2]
s
[3,2]
v1
[3,4]
v4
[3,1]
v2
d =3
[1,2]
[1,4]
t
[3,4]
[3,2]
v3
[2,2]
i
[c(ij),u(ij)]
j
Graph foundations of
network modelling 81
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
4. Maximum flow problem
5. Minimum cost flow problem
5.1
5.2
5.3
5.4
5.5
Introduction
Problem definition
Residual network
Busacker-Gowen algorithm
Several supply/demand vertices
6. Extension: multi-commodity flow
problems
Graph foundations of
network modelling 82
Several supply/demand vertices
Example :
Optimisation model :
minimize
 c( a ) f ( a )
s1
[c(ij),u(ij)]
i
[5,2]
t1
b(s1) = +5
a A
subject to
 f (a ) -
a I ( v )
[1,1]
 f (a ) = b(v) , " v V
a I '( v )
s2
0  f ( a )  u( a ) , " a  A
Min cost flow algorithm :
[0,5]
super-supply
s
s1
[1,1]
[0,2]
[1,1]
s2
[5,2]
[1,1]
[2,3]
[1,1]
t2
[2,3]
t1
b(s2) = +2
b(t1) = -4
b(t2) = -3
[0,4]
t
[1,1]
t2
j
super-demand
[0,3]
demand = 5+2 = 4+3 = 7
Graph foundations of
network modelling 83
Outline
0. Introduction
1. Definition and notations
2. Minimum spanning tree problem
3. Shortest path problem
4. Maximum flow problem
5. Minimum cost flow problem
6. Extension: multi-commodity flow problems
Graph foundations of
network modelling 84
The concept ‘commodity’
A commodity = one uniform product
Sections 3, 4 and 5 : single-commodity flow problems
e.g. oil / electricity distribution system
This section : multi-commodity flow problems
e.g. telephone network
Several multi-commodity problems : natural extensions of
the single-commodity problems
e.g. multi-commodity minimum cost flow problem
Graph foundations of
network modelling 85
Multi-commodity min cost flow problem
Optimisation model for the multi-commodity min cost flow problem :
 c( a )  f k ( a )
minimize
a A
subject to
k K
weighted capacitated digraph
 f k (a ) -  f k (a ) = bk (v)
a I ( v )
0
, "v V , "k  K
a I '( v )
 f k ( a )  u( a )
, "a  A
f1
k K
Distinction between two problems :
1
• fk(a) integer : integer MCMCF problem
• fk(a) real : linear MCMCF problem
Capacity of arc = 1
f1=f2=f3=1
integer MCMCF : cost = 202
linear MCMCF : cost = 153
f2
0.5
1
1
0.5
1 cost=100
2
f1
f3
3
f2
cost=1
f3
Graph foundations of
network modelling 86
Application: IP routing vs. MPLS Traffic Eng.
Problem: demands for traffic between every pair of IP(/MPLS) routers
IP:
only shortest paths allowed
 algorithm of Dijkstra
A
B
E
D
C
IP/MPLS:
explicit routes possible (‘traffic engineering’)
mathematical formulation: multi-commodity flow problem
very refined granularity
(one IP packet = some bytes  typical arc capacity = 100s Mbit/s)
flow variable has an almost continuous nature
 real fk(a) variables allowed
Graph foundations of
network modelling 87
Limitations of graph algorithms
Example of multi-commodity flow problems:
• distinction between real and integer problem arises
• no efficient graph algorithms as solution method
More in general:
• graph algorithms for ‘easy’ problems
• or parts of ‘difficult’ problems (e.g. network design)
 next chapter:
• general optimisation techniques
• real vs. integer
Graph foundations of
network modelling 88
References
[1] N. Christofedes: Graph Theory: an Algorithmic Approach
(Academic Press, New York, 1975).
[2] G. Chartrand, O. Oellermann: Applied and Algorithmic Graph
Theory (McGraw-Hill, New York, 1993).
[3] R.K.Ahuja, T.L.Magnanti, J.B.Orlin, Network Flows (PrenticeHall, New Jersey, 1993).
[4] M.O.Ball, T.L.Magnanti, C.L.Monma, G.L.Nemhauser, Handbooks
in Operations Research and Management Science, vol. 7, Network
Models (North-Holland, Amsterdam, 1995).
Graph foundations of
network modelling 89