Virtual Private Network Layout
A proof of the tree conjecture on a ring network
Leen Stougie
Eindhoven University of Technology (TUE)
&
CWI, Amsterdam
http://www.win.tue.nl/math/bs/spor/2004

15.pdf
Input to the VPN problem
•
Undirected graph
G=(V,E)
•
Subset of the vertices
W
µ
V
(terminals)
•
Communication bounds on the terminals
b(i)
for all
i
2
W
•
Unit capacity costs on the edges
c(e)
for
all
e
2
E
Communication bounds and scenarios
•
b(i)
is bound on total of incoming and outgoing
communication of node
v
(
symmetric VPN
)
•
A valid demand scenario is symmetric matrix
D=(d
ik
)
ik
2
W
with
d
ii
=0
satisfying
d
ik
¸
0
8
i,k
2
W
and
k
2
W
d
ik
∙
b(i)
8
i
2
W
•
D
is the set of all valid scenarios
VPN Robust optimization
•
Select for each pair
i,k
2
W
a path for
communication
•
Reserve enough capacity on the edges E
•
All demand in every valid communication
scenario
D
2
D
can be routed on the selected
paths
•
The total cost of reserving capacity is minimum
•
The paths are to be selected before seeing
any communication scenario
Routing variations of VPN
•
SPR (
Single path routing
)
For each pair
i,k
2
W
exactly one path
P
ik
µ
E
•
TTR (
Terminal tree routing
)
SPR with for each
i
2
W
,
[
k
2
W
P
ik
is a tree in
G
•
TR (
Tree routing
)
SPR with
[
i,k
2
W
P
ik
is a tree in
G
•
MPR (
Multi

path routing
)
For each pair
i,k
2
W
for each path
P
between
i
and
k
,
specify fraction of communication using
P
Relation between the variations
•
Lemma:
OPT(MPR)
∙
OPT(SPR)
∙
OPT(TTR)
∙
OPT(TR)
Proof:
SPR is the MPR problem with the extra restriction
that all fractions must be 0 or 1.
The other inequalities are similarly trivial.
The open VPN

problem
•
Conjecture 1:
SPR
2
P (polynomially solvable)
•
Conjecture 2:
OPT(SPR)=OPT(TR)
•
Conjecture 3:
OPT(MPR)=OPT(TR)
What do we know about VPN?
•
TR
2
P
Kumar et al. 2002
•
OPT(TR)= OPT(TTR)
Gupta et al. 2001
•
OPT(TR)
∙
2OPT(MPR)
Gupta et al. 2001
•
MPR
2
P
Erlebach and Ruegg 2004, Altin et al. 2004,
Hurkens et al. 2004
The asymmetric VPN
b
+
(v)
outgoing communication bound
b

(v)
incoming communication bound
•
TR is NP

hard
Gupta et al. 2001
•
TR
2
P if
v
2
W
b

(v)=
v
2
W
b
+
(v)
Italiano et al. 2002
•
MPR
2
P
Erlebach and Ruegg 2004, Altin et al. 2004, Hurkens et al. 2004
•
Constant Aprroximation ratios for SPR
Gupta et al. 2001, Eisenbrandt et al. 2005 (randomized)
Conjecture 3 is true:
•
If
G
is a tree (trivial)
•
If
G
is
K
4
•
If
G
is a cycle !!!!
•
If
G
is a 1

sum of graphs for which
Conjecture 3 is true
Path

formulation of VPN
P
ik
set of paths in
G
between
i
and
k
P
set of all paths in
G
For each path p in G we define
x
p
For all
i
and
k
2
W,
p
2
P
ik
x
p
=1
•
SPR:
x
p
2
{0,1}
8
p
2
P
•
MPR:
0
∙
x
p
∙
1
8
p
2
P
The capacity problem
•
Given selected paths: given values for
x(p)
•
Problem: find capacities on edges
z(e)
8
e
2
E
•
e
p
=1
if
e
2
P
and
0
otherwise
Dual of the capacity finding problem
Path

formulation of SPR
•
SPR: Find
x(p)
minimizing
e
2
E
c
e
z
e
Path

formulation of MPR
•
MPR: SPR with
x(p)
¸
0
i.o.
x(p)
2
{0,1}
Dual of the Path

formulation of MPR
Dual

MPR
MPR and TR
•
OPT(MPR)
∙
OPT(TR)
•
Weak duality: any feasible
(
,
)
has
ik
∙
OPT(MPR)
•
Conjecture 3:
OPT(MPR)=OPT(TR)
•
Conjecture 3:
OPT(TR)=Optimal solution
value of the dual of MPR
Optimal solution of TR (1)
Notation
b(U)=
v
2
U
b(v)
Take tree
T
Each
e
2
T
is cut in
T
splitting
V
in
L(e)
and
R(e)
Direct
e
to minimum of
b(L(e))
and
b(R(e))
•
There is a unique vertex
r
with indegree
0
, root
•
Cost of
T
:
e
min{b(L
e
),b(R
e
)} c(e)
•
The minimum cost tree with
r
as the root is the shortest
path tree from
r
in
G
w.r.t. length function
c
•
OPT(TR) can be found in polynomial time
Optimal solution of TR (2)
Let
d
G
(u,v)
the distance between
u
and
v
in
G
w.r.t. length
function
c
The cost of optimal tree
T
is given by
v
b(v)
d
T
(r,v)
for some root vertex
r
.
Moreover, it is bounded from below by
v
b(v)
d
G
(r,v)
.
Clearly it is bounded from above by
v
b(v)
d
T
(u,v) forall u
2
V
Compute shortest path tree rooted at
u
for all
u
2
V
and select
the one with minumum cost solves OPT(TR) in polynomial time
Conjecture 3 true for the cycle
Lemma:
If Conjecture 3 is true for any cycle with:

W=V

b(v)=1
8
v
2
V

V
is even
Then Conjecture 3 is true for any cycle
Theorem:
Conjecture 3 is true for any even cycle
with the above three properties
The even cycle (1)
•
Vertices
0,1,2,...,2n

1
•
Edges
e
1
,e
2
,...,e
2n
•
Cost of tree by deleting edge
e
k
:
•
(using
e
min{b(L
e
),b(R
e
)} c(e)
)
•
We show there exist a dual solution with
value equal to
min
e
k
The even cycle (2)
•
MPR

dual restricitions for even cycle with
b(v)=1
•
Only two possible paths between each pair of
vertices
The even cycle (3)
The Tool Lemma
•
The Tool Lemma:

Let
G=(V,E)
even circuit

b
´
1
.

F
µ
E, F
;
Then there exist
:E
!
R
+
,
not equal
0
, and
K
such that
•
support(
)
µ
F
•
8
f
2
F: K=C(f;
)=min
e
2
E
C(e;
)
•
There is a dual solution
(
,
)
with value
K
for the MPR

dual problem with cost function
The even cycle (4)
Part of Proof of Tool Lemma
•
Proof:
By induction on
F

F=1
(easy):
F={e
k
}

Take
k
=1
and
i
=0
8
i
k

Clearly,
min
e
2
E
C(e;
)=C(e
k
;
)=0

A feasible dual solution with value
0
is
e
ih
=0
,
ih
=0
8
e
2
E
8
i,h
2
V
The even cycle (5)
Part of Proof of Tool Lemma
•
Proof (continued):
F>1

Case (i): There is a k such that
e
k
2
F
and its
opposite edge
e
k+n
2
F

(
in figure read e
k
=a and e
k+n
=b
)
•
Choose
k
=
k+n
=1
and
i
=0
8
i
k,n+k
)
C(e;
)=n
8
e
2
E
•
Choose
•
Verify that
ij
=n
The even cycle (6)
•
Theorem:
Let
G=(V,E)
be an even circuit,
c: E
!
R
+
and
b(v)=1
8
v
2
V
. Then the cost of
an optimal tree solution equals the value
of an optimal dual solution.
•
Proof:
An inductive primal

dual argument
using the Tool Lemma.
(By request on the blackboard)
Postlude
•
OPT(MPR)=OPT(TR) for any graph?
•
SPR polynomially solvable for any graph?
•
Proof for the cycle is complicated!
•
Is there an easier proof for the cycle?
•
The crucial insight?
•
Complexity of the non

robust MPR

problem is also open!
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