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
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
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
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
Introduction Definition and Motivation 4 / 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
Introduction Definition and Motivation 4 / 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
Introduction Definition and Motivation 4 / 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
Introduction Definition and Motivation 4 / 29
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]
Introduction Definition and Motivation 5 / 29
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]
Introduction Definition and Motivation 5 / 29
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
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
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 }
max{∑
i∈I αi | I ⊂ {1, . . . , n},∑
i∈I wi ≤W } = νmax(G )− νmaxW (G )
Introduction Complexity results 6 / 29
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 }
max{∑
i∈I αi | I ⊂ {1, . . . , n},∑
i∈I wi ≤W } = νmax(G )− νmaxW (G )
Is network interdiction even strongly NP-complete?
Introduction Complexity results 6 / 29
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
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
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.
Introduction Current state of the art 9 / 29
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}
Introduction Current state of the art 10 / 29
Reduction to multi-objective shortest pathproblem (MOSP)
νmaxB (G ) = min{λ′(P ′) | P ′ path from sD to tD in G ′, c ′(P ′) ≤ B}
Introduction Current state of the art 11 / 29
General planar case (with a single source & sink)
Correspondence
s-t cuts in G ↔ counterclockwise s-t separating circuits
Introduction Current state of the art 12 / 29
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
Introduction Current state of the art 13 / 29
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
Introduction Current state of the art 13 / 29
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
Introduction Current state of the art 13 / 29
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.
Introduction Current state of the art 14 / 29
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.
Introduction Current state of the art 15 / 29
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.
Introduction Current state of the art 15 / 29
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
Generalizing s-t cuts
Extensions to planar network interdiction Vertex interdiction and vertex capacities 16 / 29
Generalizing s-t cuts
Extensions to planar network interdiction Vertex interdiction and vertex capacities 16 / 29
Generalizing s-t cuts
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 ).
Extensions to planar network interdiction Vertex interdiction and vertex capacities 16 / 29
Generalizing s-t cuts
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}
Extensions to planar network interdiction Vertex interdiction and vertex capacities 16 / 29
Adapting the dual network
Extensions to planar network interdiction Vertex interdiction and vertex capacities 17 / 29
Adapting the dual network
Extensions to planar network interdiction Vertex interdiction and vertex capacities 17 / 29
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 ∗
Extensions to planar network interdiction Vertex interdiction and vertex capacities 17 / 29
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 ∗
Extensions to planar network interdiction Vertex interdiction and vertex capacities 17 / 29
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 ∗
Extensions to planar network interdiction Vertex interdiction and vertex capacities 17 / 29
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 .
Extensions to planar network interdiction Vertex interdiction and vertex capacities 18 / 29
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.
Extensions to planar network interdiction Vertex interdiction and vertex capacities 19 / 29
Vertex capacities
Vertex capacities can easily be included into the model by a slightmodification of the extended dual graph.
Extensions to planar network interdiction Vertex interdiction and vertex capacities 20 / 29
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
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
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.
Extensions to planar network interdiction Multiple sources and sinks 22 / 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.
Extensions to planar network interdiction Multiple sources and sinks 22 / 29
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.
Extensions to planar network interdiction Multiple sources and sinks 23 / 29
1. Passage to interd. problem on circulations
Extensions to planar network interdiction Multiple sources and sinks 24 / 29
1. Passage to interd. problem on circulations
Extensions to planar network interdiction Multiple sources and sinks 24 / 29
1. Passage to interd. problem on circulations
u and c are extensions of u and c with c(e) =∞∀e ∈ T .
Extensions to planar network interdiction Multiple sources and sinks 24 / 29
1. Passage to interd. problem on circulations
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.
Extensions to planar network interdiction Multiple sources and sinks 24 / 29
2a. Incorporating lower bounds into the dual
Theorem ([Miller and Naor, 1995])
G admits a valid circulation. ⇔ G∗ contains no negative circuit.
Extensions to planar network interdiction Multiple sources and sinks 25 / 29
2a. Incorporating lower bounds into the dual
Theorem ([Miller and Naor, 1995])
G admits a valid circulation. ⇔ G∗ contains no negative circuit.
Extensions to planar network interdiction Multiple sources and sinks 25 / 29
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.
Extensions to planar network interdiction Multiple sources and sinks 26 / 29
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
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
Complexity for NFIP with mult. sources/sinks
Is network interdiction on planar graphs with multiplesources and sinks strongly NP-complete?
We do not know.
Extensions to planar network interdiction Complexity revisited 27 / 29
Complexity for NFIP with mult. sources/sinks
Is network interdiction on planar graphs with multiplesources and sinks strongly NP-complete?
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).
Extensions to planar network interdiction Complexity revisited 27 / 29
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 ′)
Extensions to planar network interdiction Complexity revisited 28 / 29
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 ′)
Extensions to planar network interdiction Complexity revisited 28 / 29
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