network flow interdiction on planar graphs...network flow interdiction on planar graphs rico...

Network Flow Interdiction on Planar Graphs

Rico ZenklusenInstitute for Operations Research, D–MATH, ETH [email protected]

Optimization and Applications Seminar, Zurich, March 10, 2008

Outline

1 IntroductionDefinition and MotivationComplexity resultsCurrent state of the art

2 Extensions to planar network interdictionVertex interdiction and vertex capacitiesNetwork flow security with multiple sources and sinksFinal thoughts on complexity of planar network interdiction

3 Conclusions

Outline 2 / 29

Outline



3 Conclusions

Robustness of flow networks

How sensitive is the value of a maximum flow in anetwork with respect to failures of arcs?

Nature of arc failures

1 Random failure → network flow reliability

Generalization of the s-t reliability problem#P-complete problemsTypically, Monte-Carlo methods are used to get estimates ofinteresting probabilities

2 Worst-case failure → network flow interdiction

Introduction Definition and Motivation 3 / 29

Network flow interdiction

Input: • Directed network G = (V ,E , u, c) with capacitiesu : E → N and interdiction costs c : V ∪ E → N ∪∞

• Fixed budget B ∈ N

Output: νmaxB (G ) := min{νmax(G \ R) | R ⊂ V ∪ E , c(R) ≤ B}

νmax(G) :=value of max flow in G

νmax(G ) = νmax0 (G ) = 10

B = 5→ νmaxB (G ) = 4


Network flow interdiction

Network interdiction models in scientific literature

Drug interdiction [Wood, 1993]

Military planning [Ghare, Montgomery, and Turner, 1971]

Protecting electric power grids against terrorist attacks [Salmeron,Wood, and Baldick, 2004]

Hospital infection control [Assimakopoulos, 1987]


Outline



3 Conclusions

Simple NP-completeness proofReduction from Knapsack Problem.

Input: n items with volumes {w1, . . . ,wn} and utilities {α1, . . . , αn}Output: max{

∑i∈I αi | I ⊂ {1, . . . , n},

∑i∈I wi ≤W }

Introduction Complexity results 6 / 29



∑i∈I αi | I ⊂ {1, . . . , n},

∑i∈I wi ≤W }

max{∑

i∈I αi | I ⊂ {1, . . . , n},∑

i∈I wi ≤W } = νmax(G )− νmaxW (G )




∑i∈I αi | I ⊂ {1, . . . , n},

∑i∈I wi ≤W }

max{∑

i∈I αi | I ⊂ {1, . . . , n},∑

i∈I wi ≤W } = νmax(G )− νmaxW (G )

Is network interdiction even strongly NP-complete?


Strong NP-completeness ([Wood, 1993] simplified)

Reduction from Max Clique.

∃ clique C in G with size k ⇔ νmax(G ′)− νmaxk (G ′) =

(k2

)Introduction Complexity results 7 / 29

Outline



3 Conclusions

Some progress on planar graphs

On planar networks, progresses were achieved by transforming thenetwork interdiction problem to the planar dual.

Pseudo-polynomial algorithm when the following conditions aresatisfied simultaneously ([Phillips, 1993]):

planar (undirected) network

single source and sink

no vertex removals

Introduction Current state of the art 8 / 29

s-t planar graphs

Correspondence

Elementary s-t cuts in G ↔ paths from sD to tD in G∗

Value of cut equals dual length (λ∗) of corresponding dual path.


Pseudo-polyn. algorithm for s-t planar graphs(Translation of the network interdiction problem onto the dual)

Definition (Reduced length with respect to B)

Let U∗ ⊂ E ∗.

λ∗B(U∗) = minX∗⊂E∗

{λ∗(U∗ \ X ∗) | c∗(X ∗) ≤ B}

Theorem

νmaxB (G ) = min{λ∗B(P∗) | P∗ path from sD to tD}


Reduction to multi-objective shortest pathproblem (MOSP)

νmaxB (G ) = min{λ′(P ′) | P ′ path from sD to tD in G ′, c ′(P ′) ≤ B}


General planar case (with a single source & sink)

Correspondence

s-t cuts in G ↔ counterclockwise s-t separating circuits


Characterizing countercl.w. s-t sep. circuits

P: path from s to t in G

PD = {eD | e ∈ P}PD

R = {eDR | e ∈ P}

Definition (Parity w.r.t. P)

p∗P(U∗) = |U∗ ∩ PD | − |U∗ ∩ PDR |

TheoremLet C∗ be a circuit in G∗.

C∗ is counterclockwise s-t sep.⇔

p∗P(C∗) = 1


Transformation to MOSP problem

νmaxB (G ) = min{λ∗B(C ∗) | C ∗ circuit in G ∗, p∗P(C ∗) = 1}

Transformation is done as in the s-t planar case with an additionalobjective: parity.

→ Again, the corresponding MOSP problem can be solved inpseudo-polynomial time by dynamic programming.


Restrictions of current pseudo-poly. algorithms(apart from planarity of the underlying graph)

Vertex capacities cannot be modeled

Vertex interdiction is not allowed

Bound to a single source and single sink

Vertex interdiction and vertex capacities are typically modeled bydoubling the vertices.

Multiple sources and sinks can be reduced to a single source & sink byintroduction of a supersource and supersink.

→ However, these constructions destroy planarity.


Outline



3 Conclusions

Generalizing s-t cuts

Extensions to planar network interdiction Vertex interdiction and vertex capacities 16 / 29


Definition (s-t separating set)

Q ⊂ V ∪ E is an s-t separating set (in G ) if there is no path from s to t inG \ Q. Furthermore, the reduced value of Q is defined by

uB(Q) := min{u(Q \ X ) | X ⊂ Q, c(X ) ≤ B}(convention: u(v) =∞∀v ∈ V ).



Definition (s-t separating set)

Q ⊂ V ∪ E is an s-t separating set (in G ) if there is no path from s to t inG \ Q. Furthermore, the reduced value of Q is defined by

uB(Q) := min{u(Q \ X ) | X ⊂ Q, c(X ) ≤ B}(convention: u(v) =∞∀v ∈ V ).

νmaxB (G ) = min{uB(Q) | Q s-t separating set in G}


Adapting the dual network


Adapting the dual network

Correspondence(between s-t sep. sets in G and countercl.w. s-t sep. circuits in G )

Q −→ C ∗(Q)

Q(C ∗) ←− C ∗


Correspondence between reduced values

G∗ = (V ∗ = V ∗ ∪ V , E∗ = E∗ ∪ E , λ∗, c∗, p∗P) where λ∗, c∗ and p∗P areextensions of λ∗, c∗ and p∗P .


Correspondence between reduced values (2)

Relations between G and G ∗

i) ∀ Q s-t separating sets in G

umaxB (Q) ≥ λ∗B(C ∗(Q)) .

ii) ∀ C ∗ counterclockwise s-t separating circuit in G ∗

umax(G \ Q(C ∗)) ≤ λ∗B(C ∗)

⇒ The problem can be solved as in the case without vertexinterdiction by transformation to a MOSP.


Vertex capacities

Vertex capacities can easily be included into the model by a slightmodification of the extended dual graph.


Outline



3 Conclusions

The network flow security problem(To simplify explanations we consider the case without vertex removal.)

Input: • Interdiction network G = (V ,E , u, c)• Sources S ⊂ V , sinks T ⊂ V \ S• Demand d : V −→ Z with −d(S) = d(T )

(d(s) < 0 ∀s ∈ S , d(t) > 0 ∀t ∈ T )

Output: min{B | νmaxB (G ) < νmax(G )}

→ When dealing with unit interdiction cost, the network flow securityproblem corresponds to determining if a network is n − k secure.

Extensions to planar network interdiction Multiple sources and sinks 21 / 29

Relation with network interdiction

The network flow security problem (NFSP) and single source & sinknetwork flow interdiction problem (SSSNFIP) can easily be reduced to eachother on general (not necessarily planar) graphs.

NFSP→SSSNFIP: Binary search over budget.SSSNFIP→NFSP: Binary search over capacity of the sink.

However on planar graphs no poly. reduction NFSP → SSSNFIP is known.

On planar networks NFSP can be seen as a generalization of SSSNFIP.


Pseudo-polynomial algorithm for planar NFSP

1 Transform the problem into a interdiction problem on flowcirculations by sending flow from the sources to the sinks onartificial arcs.

2 Reformulate the problem on a dual network that allows toincorporate lower bounds on capacities and transform it to aMOSP.


1. Passage to interd. problem on circulations



u and c are extensions of u and c with c(e) =∞∀e ∈ T .



u and c are extensions of u and c with c(e) =∞∀e ∈ T .

For every interdiction set R ⊂ E we have:There is a saturating flow in G \ R ⇔ There is a circulation in G \ R.


2a. Incorporating lower bounds into the dual

Theorem ([Miller and Naor, 1995])

G admits a valid circulation. ⇔ G∗ contains no negative circuit.


2b. Transformation to MOSP

The theorem of Miller & Naor can easily be extended to include thepossibility of interdiction.

Theorem

νmaxB (G ) < 0⇔ ∃ circuit C∗ in G∗ such that λ∗B(C∗) < 0

⇒ Finding a circuit with negative reduced length in G∗ can be transformedinto a MOSP similar to the previous problems.


Outline



3 Conclusions

Complexity for NFIP with mult. sources/sinks

Is network interdiction on planar graphs with multiplesources and sinks strongly NP-complete?

Extensions to planar network interdiction Complexity revisited 27 / 29



We do not know.




We do not know.

But it is at least as difficult as finding dense subgraphs of planargraphs (whose complexity is also a long standing open problem).


Reducing k-densest subgraph problem to NFIP

k-densest subgraph problem on planar graphs:

Input: Undirected planar graph G = (V ,E ), k ∈ NOutput: max{#edges in G [V ′] | V ′ ⊂ V , |V ′| = k}

max{#edges in G [V ′] | V ′ ⊂ V , |V ′| = k} = νmax(G ′)− νmaxk (G ′)


Conclusions

Network interdiction is strongly NP-complete. Pseudo-polynomialalgorithms were only available for (undirected) planar graphs with asingle source & sink and without vertex interdiction.

Pseudo-polynomial algorithms on directed planar graphs for thefollowing extensions were presented:

Vertex interdiction & vertex capacitiesMultiple sources and sinks in the context of network security.

Hardness-result/algorithm is missing for network interdiction on planargraphs with multiple sources and sinks.

The problem is at least as hard as the k-densest subgraph problemon planar graphs.

Conclusions 29 / 29

References I

N. Assimakopoulos. A network interdiction model for hospital infectioncontrol. Computers in biology and medicine, 17(6):413–422, 1987.

P. M. Ghare, D. C. Montgomery, and W. C. Turner. Optimalinterdiction policy for a flow network. Naval Research LogisticsQuarterly, 18:37–45, 1971.

G. L. Miller and J. Naor. Flow in planar graphs with multiple sourcesand sinks. SIAM J. Comput., 24(5):1002–1017, 1995. ISSN0097-5397. doi: http://dx.doi.org/10.1137/S0097539789162997.

Conclusions 30 / 29

References II

C. A. Phillips. The network inhibition problem. In STOC ’93:Proceedings of the twenty-fifth annual ACM symposium on Theoryof computing, pages 776–785, New York, NY, USA, 1993. ACMPress. ISBN 0-89791-591-7. doi:http://doi.acm.org/10.1145/167088.167286.

J. Salmeron, K. Wood, and R. Baldick. Analysis of electric gridsecurity under terrorist thread. IEEE Transaction on Power Systems,19(2):905–912, 2004.

R. K. Wood. Deterministic network interdiction. Mathematical andComputer Modeling, 17(2):1–18, 1993.

Conclusions 31 / 29

network flow interdiction on planar graphs...network flow interdiction on planar graphs rico...

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