1
Multiconstrained QoS Routing: Simple
Approximations to Hard Problems
Guoliang (Larry) Xue
Arizona State University
Research Supported by
ARO
and
NSF
Collaborators: W. Zhang, J. Tang, A. Sen, and
K. Thulasiraman
2
Outline/Progress of the Talk
Problem Definitions
Related Works
Simple K

Approximation Algorithms
Faster Approximation Schemes
Conclusions
3
Multi

Constrained QoS Routing
Given a network where each link
e
has a
cost
c(e)
and a
delay
d(e),
we are interested in finding a source

destination path whose
cost is within a given cost
tolerance
C
and whose
delay is within a given delay
tolerance
D
.
This problem is NP

hard. There are many
heuristic
algorithms which have
no performance guarantees
, and
sophisticated
approximation schemes
which are
too
complicated
for protocol implementation
.
We have designed the
fastest approximation schemes
, as
well as
very simple hop

by

hop routing algorithms
that
have good performance guarantees.
4
Multi

Constrained QoS Routing
We study the
general
problem where there are
K
QoS
parameters
,
for any constant
K
≥2
.
We are given an undirected graph G(V, E) where each edge
e
E is associated with K nonnegative weights
1
(e),
2
(e),
…,
K
(e). We are also given a
source s
and
destination t
,
and K positive constants W
1
, …, W
K
.
The multi

constrained QoS routing problem asks for
an s
—
t
path p
such that
k
(p) ≤ W
k
, for k=1, 2, …, K.
For simplicity
, we assume K=2 for the most part of this talk.
In this case, we will talk about
cost
and
delay
.
5
Illustration of the Problem (C=W
1
, D=W
2
)
s
x
y
z
(2, 5)
(12, 5)
(10, 0)
K
= 2
W
1
= 16,
W
2
= 8
The shortest path with regard to the
1
st
edge weight
is
(s, z)
(14, 1)
The shortest path with regard to the
2
nd
edge weight
is
(s, y, z)
Neither of them is a feasible solution
!
Path
(s, x, y, z) is a feasible path.
6
Outline/Progress of the Talk
Problem Definitions
Related Works
Simple K

Approximation Algorithms
Faster Approximation Schemes
Conclusions
7
Related Works
J.M. Jaffe,
Algorithms for finding paths with multiple constraints
,
Networks
, 1984.
S. Chen and K. Nahrstedt,
On finding multi

constrained paths
,
IEEE
International Conference on Communications
, 1998.
X. Yuan,
Heuristic algorithms for multiconstrained quality of service
routing
,
IEEE/ACM Transactions on Networking
, 2002.
R. Hassin,
Approximation schemes for the restricted shortest path
problems
,
Mathematics of Operations Research
, 1992.
D.H. Lorenz and D. Raz,
A simple efficient approximation scheme for the
restricted shortest path problem
,
Operations Research Letters
, 2001.
G. Xue, A. Sen, W. Zhang, J. Tang, K. Thulasiraman;
Finding a path
subject to many additive QoS constraints
;
IEEE/ACM Transactions on
Networking
,
2007.
G. Xue, W. Zhang, J. Tang, K. Thulasiraman;
Polynomial time
approximation algorithms for multi

constrained QoS routing
;
IEEE/ACM
Transactions on Networking
,
2008.
8
Related Works
G. Xue;
Minimum cost QoS multicast and unicast routing in
communication networks
;
IPCCC’2000/IEEE Transactions on
Communications,
2003.
A. Junttner et al.,
Lagrange relaxation based method for the QoS routing
problems
,
IEEE INFOCOM,
2001.
A. Goel et al.,
Efficient computation of delay

sensitive routes from one
source to all destinations
,
IEEE INFOCOM,
2001.
T. Korkmaz and M. Krunz,
A randomized algorithm for finding a path
subject to multiple QoS requirements
,
Computer Networks
, 2001.
P. Van Mieghem et al.,
Concepts of exact QoS routing algorithms
,
IEEE/ACM Transactions on Networking
, 2004.
F.A. Kuipers et al.,
A comparison of exact and eps

approximation
algorithms for constrained routing
,
IFIP NETWORKING
, 2006.
A. Orda and A. Sprintson.,
Efficient algorithms for computing disjoint QoS
paths
,
IEEE INFOCOM
, 2004.
9
Outline/Progress of the Talk
Problem Definitions
Related Works
Simple K

Approximation Algorithms
Faster Approximation Schemes
Conclusions
10
A
Simple
Idea
The
decision problem
is to find a path p such that c(p)
≤
C
and d(p)
≤
D.
The
optimization problem
is to find a path p such that
max {c(p)/C, d(p)/D}
is minimized.
Define
l(p) = max {c(p)/C, d(p)/D} as a new path length.
The original problem has a feasible solution if and only if
there is a path p such that l(p)
≤
1.
The optimization problem is NP

hard as well.
The Idea
: For each link e,
define
a new link weight
w(e) = max{c(e)/C, d(e)/D}.
The shortest path with respect to w(e) can be computed
easily, and is
within a factor of 2 from the optimal solution
.
11
Illustration of the Concepts (C=W
1
, D=W
2
)
s
x
y
z
(2, 5)
(12, 5)
(10, 0)
K
= 2
W
1
= 16,
W
2
= 8
The shortest path with regard to the
1
st
edge weight
is
(s, z), l(p)=
20/8
.
(14, 1)
The shortest path with regard to the
2
nd
edge weight
is
(s, y, z), l(p)=
11/8
.
Neither of them is a feasible/optimal solution !
The optimal path is
(s, x, y, z), l(p)=
7/8
12
A Simple 2

Approximation Algorithm
s
x
y
z
(2, 5)
(12, 5)
(10, 0)
K
= 2
W
1
= 16,
W
2
= 8
(14, 1)
The shortest path with regard to the new edge weight is
(s, y, z)
whose path length is 11/8.
This path has a length that is guaranteed to be within a factor of 2 from the optimal value
.
In this case, we have 11/8
≤
2
×
7/8.
(2/16,
5/8
)
5/8
14/16
12/16
10/16
13
A
Better
Greedy 2

Approximation Algorithm
s
x
y
z
(2, 5)
(12, 5)
(10, 0)
K
= 2
W
1
= 16,
W
2
= 8
(14, 1)
The path found by Greedy is
(s, x, z)
with path length 1
[0,0]
[2/16,
5/8
]
A path from
s
to
x
with path weights [2/16, 5/8] is stored at node
x
. The
path length
is 5/8
[12/16,
20/8
]
[
12/16
, 5/8]
The path at node
x
is chosen because it has the minimum
path length
[4/16,
7/8
]
[
16/16
, 6/8]
The path at node
y
is chosen because it has the minimum
path length
among the unmarked nodes
[
22/16
, 5/8]
The optimal solution is
(s, x, y, z)
with path length 7/8
14
Proof of Correctness
K

Approx:
The central idea used in the proof of K

Approx relies on
the following simple fact.
Let
x be a point in the K

dimensional Euclidean plane
.
Then
x
≤x
1
≤K
x
Greedy:
Greedy never violates the upper

bound on path length
used in the proof of K

Approx.
15
Numerical Results
Algorithms compared
Greedy
Previously best known
K

approximation algorithm
FPTAS for the OMCP problem
K
= 3,
W = W
1
= W
2
= W
3
Networks
well

known Internet topologies
•
ArpaNet (20 nodes and 32 edges) and ItalianNET (33 nodes, 67 edges)
randomly generated topologies
BRITE
[BRITE]
Waxman model
[WaxJSAC88]
, and have the default parameters set by BRITE
the edge weights were uniformly generated in a given range (we used the
range [1,10]).
Three scenarios
•
Infeasible W = 5
•
Tight W = 10
•
Loose W = 20
[BRITE] BRITE; http://www.cs.bu.edu/brite/.
[WaxJSAC88] B.M. Waxman; Routing of multipoint connections;
IEEE Journal on Selected Areas in Communications
; Vol. (1988).
(
ε
= 0.1)
16
On ArpaNet Topology
The number of
better
paths: path
p
1 is better than path
p
2 if l(p1) < l(p2)
For any path
p
, its
relative error
is calculated as (l(p)

l(p
OMCP
))/
l(p
OMCP
) ,
where
p
OMCP
is the path found by
OMCP
for the source

destination pair.
17
On Large Random Network Topologies
Path quality, eps = 0.1, 100 nodes, 390 links.
Scalability of the algorithms, eps=0.5.
80x314, 210x474, 140x560, 160x634.
18
Outline/Progress of the Talk
Problem Definitions
Related Works
Simple K

Approximation Algorithm
Faster Approximation Schemes
Conclusions
19
Approximation Scheme for SMCP
We are given an undirected graph G(V, E) where each edge
e
E is associated with K nonnegative weights
1
(e),
2
(e),
…,
K
(e). We are also given a
source s
and
destination t
,
and K positive constants W
1
, …, W
K
. We want to find an s

t
path p s.t max{
k
(p)/ W
k
, 1
≤k≤K} is minimized.
In a paper published in TON’2007, we designed an algorithm
that can find a
(1+
)

approximation in O(m(n/
)
K

1
) time
.
This is the first FPTAS for the general SMCP problem (K
2).
20
Approximation Scheme for SMCP
The idea follows that used by other researchers in this field.
Find an initial pair of lower and upper bounds not too far away from each
other.
Use scaling/rounding/approximate testing to refine the bounds to within a
constant factor
Compute an (1+
)

approximation.
The difference is that we got a pair of lower and upper bounds with a
constant (K) factor
in a single step
, using our K

Approx. This leads to
faster running time.
O(mn(loglogn+1/
))
O(mn/
) for K=2
.
However, the problem is
slightly different
from the DCLC problem.
None
of the constraints is enforced
. Motivation for the second TON paper.
21
Faster Approximation Schemes for OMCP
All previous approximation schemes for OMCP are based on
Initial bounds
Scaling and rounding, and approximate testing
Final solution
Hassin rounds to
floor
. Lorenz and Raz round to
floor plus
one,
and showed its advantage over that of Hassin.
A
simple combination of the two
techniques leads to an
approximation scheme that is
better than both
.
22
Basic Definitions Again
23
Decision Version
of the Basic Problem
24
A
Restricted Decision Version
(for comparison)
25
Optimization Version
of the Problem
26
The DCLC Problem (used as a subproblem)
27
MCPN:
non

negative integers
(2 weights)
28
MCPP:
positive integers
(K weights)
29
Solving MCPP in O(mC
K

1
) Time
30
Solving MCPP in O(mC
K

1
) Time
31
Illustration of the idea (acyclic graph)
32
Solving MCPN in O((m+nlogn)C) Time
33
Solving MCPN in O((m+nlogn)C) Time
34
Solving MCPN in O((m+nlogn)C) Time
35
Non

negative
rounding
and
approximate testing
36
Non

negative
rounding
and
approximate testing
37
Positive
rounding
and
approximate testing
38
Positive
rounding
and
approximate testing
39
The Power of Approximate Testing
Assume UB
2(1+
)LB
.
Set
C
=sqrt(LB
UB/(1+
)).
Run
TEST(C,
).
If TEST(C,
)
YES,
then
DCLC
<C(1+
).
Decrease UB to
C(1+
).
If TEST(C,
)
NO,
then
DCLC
>C.
Increase LB to C.
In both cases,
UB/LB
is reduced to
sqrt((1+
)(UB/LB)).
We will have UB≤2(1+
)LB, after
loglog(initial UB/LB ratio)
iterations.
40
Faster Approximation Scheme for DCLC
Use the technique of Lorenz and Raz to compute LB and UB of
DCLC
so
that
LB ≤
DCLC
≤UB≤n
LB.
This takes
O((m+nlogn)logn))
time:
logn
shortest path computations
.
Set
N
to (logn)
2
, and apply TEST
N
to refine LB and UB so that
LB≤
DCLC
≤UB≤2(1+
(logn)
2
)
LB.
This takes
O(mn)
time:
loglog(n) TEST
N
,
each requires O((m+nlogn)n/
N
) time
.
Set
P
to 1
, and apply TEST
P
to refine LB and UB so that
LB≤
DCLC
≤UB≤2(1+
1
)
LB.
This takes
O(mnlogloglogn)
time:
loglog(logn)
TEST
P
, each requires O(mn/
P
) time
.
Solve MCPP with
scaling factor
=(n

1)/(LB
).
This takes
O(mn/
)
time.
O(mn(loglogn+1/
))
O(mn(logloglogn+1/
))
41
Faster Approximation Scheme for DCLC
42
Faster Approximation Scheme for DCLC
43
Faster Approximation Scheme for DCLC
O(mn(loglogn + 1/
)) time
[Lorenz and Raz ORL’2001]
O(mn(logloglogn + 1/
)) time.
Conjecture:
O(mn/
) time both necessary and sufficient.
44
Dimension Reduction: OMCP
䑃䱃
45
(1+
⤨)

ㄩ

䅰灲潸 瑯⁏䵃倬癩v 䑃䱃
46
Faster Approximation Scheme for OMCP
47
Faster Approximation Scheme for OMCP
This is essentially O(m(n/
)
K

1
) time.
48
Faster Heuristic/Scheme for DMCP
49
Faster Heuristic/Scheme for DMCP
O(mn(n/
)
K

1
) time [Yuan TON’2002]
O(m(H/
)
K

1
) time.
50
Running Time
51
Running Time
52
Path Weight Ratios
53
Outline/Progress of the Talk
Problem Definitions
Related Works
Simple K

Approximation Algorithm
Faster Approximation Schemes
Conclusions
54
Conclusions
We know how to compute a
shortest path
. OSPF has been
proposed by IETF as an RFC.
We don’t know how to handle two or more QoS constraints
with guaranteed performance.
This is the first approach which is
both simple and
provably good
.
It is as simple as computing a shortest path.
The computed path is within a factor of K from optimal.
From the theoretical point of view, we have designed
faster FPTAS for several versions of the problem.
55
THANK YOU!
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