a graph-theoretic qos-aware vulnerability assessment for network topologies

Problem Definition and ContributionsDiscussion on Hardness

Algorithms for different network settingsSimulation

A Graph-theoretic QoS-aware Vulnerability

Assessment for Network Topologies

Ying Xuan

March 24, 2010

Ying Xuan A Graph-theoretic QoS-aware Vulnerability Assessment for Netw



Table of contents

1 Problem Definition and Contributions

2 Discussion on Hardness

3 Algorithms for different network settings

network with small amount of constraints

small networks with unrestricted constraints

two fast heuristic algorithms for general networks

4 Simulation




System Model

given a networks with a set of s − t paths;

each path satisfies some of the constraints, thus has a satisfactory score by

assigning each constraint a score.

as long as there exists a path with score higher the some threshold, the network

is still operational.

goal: at least home many node/link failures will make the network not

operational, i.e. no such satisfiable path exists, and QoS routing service is not

available from s to t.




Graph Model

directed graph G(V , E), source s, destination t;

additive weight vector for each edge ej = (u, v) ∈ E :(

wj1,w

j2, · · · ,w

jm

)

.

weight vector for each path:(

∑

ej∈p wj1,

∑

ej∈p wj2, · · · ,

∑

ej∈p wjm

)

;

constraint threshold vector (c1, c2, · · · , cm);

priority vector (λ1, λ2, · · · , λm);

we define a satisfactory score φ(p) for path p as:

φ(p) =∑

j :p∝i

λj

where p ∝ i iff wpi≤ ci




Graph Model

E

H

(2,2)

D

G

(1,1)

F

I(2,2) (3,2)

(2,1)(2,2)

(2,2) (1,1) (2,3)

(1,2)

(3,7)(1,2) BA C

Given a score threshold ρ, find the minimum number of edges whose removal makes

φ(P) = max{φ(p) | ∀s − t path p in G} ≤ ρ




Integer Program

Variables:

Xe = 1 if edge e is NOT removed in the optimal solution, 0 otherwise;

Ypi = 1 if a s − t path p has wpi≤ ci , 0 otherwise;

a large constant ǫ = maxi{∑

e∈E wei }.




Integer Program

min∑

e∈E

(1− Xe)

s.t.∑m

i=1 Ypiλi ≤ ρ, ∀s − t path p∑

e∈p wei≤ ǫ− Ypi ǫ +

∑

e∈p(1− Xe)ǫ + ci , ∀i ,p∑

e∈p wei > 1− Ypi ǫ−

∑

e∈p(1 − Xe )ǫ + ci , ∀i , p

Xe ∈ [0, 1]

Ypi ∈ [0, 1]

Figure: Integer Programming Formulation




Main Contributions

provide the first graph-theoretic QoS-ware vulnerability assessment method;

abstract the assessment problem as a graph optimization problem and study its

hardness;

present exact solutions to the problem in two practical cases and several

heuristics for general cases.




decide version

Given a set of edges deleted, does there exist a path P satisfying φ(P) ≤ ρ in the

remaining graph.

Definition (QoS-SP)

Given a graph with an m-dimension constraint vector, find a path P such that

φ(P) = maxp=[s,··· ,t]∈G

φ(p)

Definition (MCP)

Given m constraints Ci , the problem is to find a path P from a source node s to a

destination node t such that

wi (P) ,∑

(u,v)∈P

wi (u, v) ≤ Ci

for all i ∈ [1, m]Ying Xuan A Graph-theoretic QoS-aware Vulnerability Assessment for Netw



NP-hard

Lemma

QoS-SP problem is NP-Complete.

Proof.

Consider the decision version of QoS-SP: given a positive number k, decide if there

exists a path P with φ(P) ≥ k. This proof is straightforwardly by reduction from MCP

problem by letting k =∑m

i=1 λi . Then if there exists a solution for the QoS-SP

problem, then the path is a feasible path to MCP, otherwise, MCP has no solution.

Since MCP is NP-hard, the proof completes.




network with small amount of constraintssmall networks with unrestricted constraintstwo fast heuristic algorithms for general networks

Exact Solution

Enumerate all (up to 2m) satisfiable constraint states;

Employ FringeMCSP [5] to find the shortest path that satisfies a specific set of

constraints;

Use revised Edmonds-Karp algorithm to remove all satisfiable augmenting path,

w.r.t a set of constraints.

Find a minimum set of edge cut.





Exact Solution

Exact Algorithm

S ← ∅;

for any maximal subset ss of M with∑

ci∈ss λi < ρ do

S ← S ∪ {M \ ss}

while S 6= ∅ do

ss ← extracted from S;

for each edge (i , j) ∈ E do

Set f (i , j) = f (j , i) = 0; Set cf (i , j) = 1 and cf (j , i) = 0;

while shortest path q that satisfies all constraints in ss can be found using

FringeMCSP do

for each edge (u, v) ∈ q do

cf (q) = min{cf (u, v) : (u, v) ∈ q}; f (u, v) = f (u, v) + cf (q);

f (v , u) = −f (u, v); cf (u, v) = c(u, v)− f (u, v);

cf (v , u) = c(v , u)− f (v , u);

all the vertices reachable from s on the residual network induces a cut T .Return the minimum cut among the cuts derived.





Exact Solution

Enumeration of satisfiable constraint states does not work.

Exact Solution:

Employ an existing all-path dynamic programming algorithm label-correcting [3]

to discovery satisfiable paths;

Use revised Edmonds-Karp algorithm [6] to return the cut.





Label-Correcting

Label-Correcting to find satisfiable paths

S ← ∅; labelSet(s) ← {(0, 0, · · · , 0, ∅)};

Q ← {s};

while Q 6= ∅ do

u ← extracted from the end of Q;

for all outgoing edges (u, v) do

labelList′

(v)← Merge(labelList(v), labelList(u), (u, v));

/*extend tuples of u to v by adding the weights of (u, v), eliminate all

dominated tuples.*/

if labelList′

(v) 6= labelList(v) then

labelList(v)← labelList′

(v), Q ← Q ∪ {v};S ← all the PO paths obtained from labelList(t);

Return q = maxp{φ(p) | p ∈ S}.





Strategies

Adopt a Relax-SAT test for estimate the existence of satisfiable path in the

remaining graph;

Adopt two greedy metrics to iteratively remove edges until the Relax-SAT test

returns Negative result.





Relax-SAT test

Relax-SAT Metric

For each edge e,

ϕ1(e) = −m

∑

i=1

wei

ci

λi

For each path p,

ϕ1(p) =∑

e∈p

ϕ(e)

Similarly, define

ϕ2(e) =m

∑

i=1

wei

ci

λi





How far from SAT

Assume λ ,∑m

i λi , β(p) , maxi (wp

ci);

Assume p satisfies a set Cs of constrains, and does not satisfy the set Cs , then

φ(p) =∑

ci∈Csλi and ϕ1(p) = −

∑

ci∈Cs

wpi

ciλi −

∑

cj∈Cs

wpj

cjλj , therefore

φ(p) ≥ ρ ⇒ ϕ1(p) ≥ −ρ − (λ − ρ)β(p);

ϕ1(p) ≥ −ρ ⇒ φ(p) ≥ ρ;

since it is hard to calculate β(p) when p is not determinate, we assert

maxϕ1(p) = −minϕ2(p) ≤ −ρ ⇒ maxφ(p) ≤ ρ;

−minϕ2(p) ≥ −ρ ⇒ maxφ(p) ≥ ρ;





Relax-SAT test

Relax-SAT test

Input: directed graph G = (V , E), constant ρ;

Output: is there a satisfiable path.

for every edge e ∈ E do

ϕ2(e)←∑m

i=1we

ici

λi ;

q ← shortest s-t path on metric ϕ2;

if ϕ2(q) > ρ then

Return NO;

if ϕ2(q) < ρ then

Return YES;





Two greedy metrics

Definition (Nonlinear Mixed Edge Metric)

ξ(e) =n

∑

i=1

(

wi (e)π(i , e)

ci

)

α

λi

where wi (e) is the i th weight on edge e, π(i , e) is the length of a s − t path, which

has the minimum weight w.r.t the i th constraint over all s − t paths containing e, and

λi is the priority parameter for the i th constraint.

It is evident that the smaller ξ(e) is, the more likely it will be included in the

QoS-optimal path. α ∈ [1, · · · , +∞) is tuned to increase the likelihood of this path to

be selected in the optimal solution.





Two greedy metrics

Definition (Betweenness)

θ(e) ,∑

i : e∈Pi

λi

where Pi refer to the shortest path w.r.t the i th constraint.




Simulation Results

To be announced.




Bibliography

Fernado A. Kuipers and Piet F. A. Van Mieghem, “Conditions that Impact the

complexity of QoS Routing”, IEEE transaction on Networking, 2005

Line B. Reinhardt and David Pisinger, “Multi-Objective and Multi-Constrained

Non-Additive Shortest Path Problems”, Technique Report, DTU Management ,

2009.

Turgay Korkmaz and Marwan Krunz, “Multi-Constrained Optimal Path

Selection”, Infocom, 2001.

P. Kh adivi, S. Samavi and T. D. Todd, “Multi-constraint QoS routing using a

new single mixed metrics”, Journal of Network and Computer Applications, 2008.

Yuxi Li, Janelee Harms, Robert Holte, “Fast Exact MultiConstraint Shortest

Path Algorithms”, ICC, 2007.

Jack Edmonds and Richard M. Karp, “Theoretical improvements in algorithmic

efficiency for network flow problems”, Journal of the ACM 19 (2): 248õ264.


a graph-theoretic qos-aware vulnerability assessment for network topologies

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