Network Flow Interdiction on Planar Graphs...Network Flow Interdiction on Planar Graphs Rico...
Transcript of 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
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