Network Flow Interdiction on Planar Graphs

Transcription

1 Network Flow Interdiction on Planar Graphs Rico Zenklusen Institute for Operations Research, D MATH, ETH Zurich rico.zenklusen@ifor.math.ethz.ch Optimization and Applications Seminar, Zurich, March 10, 2008

2 Outline 1 Introduction Definition and Motivation Complexity results Current state of the art 2 Extensions to planar network interdiction Vertex interdiction and vertex capacities Network flow security with multiple sources and sinks Final thoughts on complexity of planar network interdiction 3 Conclusions Outline 2 / 29

3 Outline 1 Introduction Definition and Motivation Complexity results Current state of the art 2 Extensions to planar network interdiction Vertex interdiction and vertex capacities Network flow security with multiple sources and sinks Final thoughts on complexity of planar network interdiction 3 Conclusions

4 Robustness of flow networks How sensitive is the value of a maximum flow in a network 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 problems Typically, Monte-Carlo methods are used to get estimates of interesting probabilities 2 Worst-case failure network flow interdiction Introduction Definition and Motivation 3 / 29

5 Robustness of flow networks How sensitive is the value of a maximum flow in a network 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 problems Typically, Monte-Carlo methods are used to get estimates of interesting probabilities 2 Worst-case failure network flow interdiction Introduction Definition and Motivation 3 / 29

6 Network flow interdiction Input: Directed network G = (V, E, u, c) with capacities u : E N and interdiction costs c : V E N Fixed budget B N Output: ν max B (G) := min{νmax (G \ R) R V E, c(r) B} ν max (G) :=value of max flow in G Introduction Definition and Motivation 4 / 29

7 Network flow interdiction Input: Directed network G = (V, E, u, c) with capacities u : E N and interdiction costs c : V E N Fixed budget B N Output: ν max B (G) := min{νmax (G \ R) R V E, c(r) B} ν max (G) :=value of max flow in G Introduction Definition and Motivation 4 / 29

8 Network flow interdiction Input: Directed network G = (V, E, u, c) with capacities u : E N and interdiction costs c : V E N Fixed budget B N Output: ν max B (G) := min{νmax (G \ R) R V E, c(r) B} ν max (G) :=value of max flow in G ν max (G) = ν max 0 (G) = 10 Introduction Definition and Motivation 4 / 29

9 Network flow interdiction Input: Directed network G = (V, E, u, c) with capacities u : E N and interdiction costs c : V E N Fixed budget B N Output: ν max B (G) := min{νmax (G \ R) R V E, c(r) B} ν max (G) :=value of max flow in G ν max (G) = ν max 0 (G) = 10 B = 5 ν max B (G) = 4 Introduction Definition and Motivation 4 / 29

10 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

11 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

12 Outline 1 Introduction Definition and Motivation Complexity results Current state of the art 2 Extensions to planar network interdiction Vertex interdiction and vertex capacities Network flow security with multiple sources and sinks Final thoughts on complexity of planar network interdiction 3 Conclusions

13 Simple NP-completeness proof Reduction from Knapsack Problem. Input: n items with volumes {w 1,..., w n } and utilities {α 1,..., α n } Output: max{ i I α i I {1,..., n}, i I w i W } Introduction Complexity results 6 / 29

14 Simple NP-completeness proof Reduction from Knapsack Problem. Input: n items with volumes {w 1,..., w n } and utilities {α 1,..., α n } Output: max{ i I α i I {1,..., n}, i I w i W } max{ i I α i I {1,..., n}, i I w i W } = ν max (G) νw max(g) Introduction Complexity results 6 / 29

15 Simple NP-completeness proof Reduction from Knapsack Problem. Input: n items with volumes {w 1,..., w n } and utilities {α 1,..., α n } Output: max{ i I α i I {1,..., n}, i I w i W } max{ i I α i I {1,..., n}, i I w i W } = ν max (G) νw max(g) Is network interdiction even strongly NP-complete? Introduction Complexity results 6 / 29

16 Strong NP-completeness ([Wood, 1993] simplified) Reduction from Max Clique. clique C in G with size k ν max (G ) ν max k (G ) = ( ) k 2 Introduction Complexity results 7 / 29

17 Outline 1 Introduction Definition and Motivation Complexity results Current state of the art 2 Extensions to planar network interdiction Vertex interdiction and vertex capacities Network flow security with multiple sources and sinks Final thoughts on complexity of planar network interdiction 3 Conclusions

18 Some progress on planar graphs On planar networks, progresses were achieved by transforming the network interdiction problem to the planar dual. Pseudo-polynomial algorithm when the following conditions are satisfied simultaneously ([Phillips, 1993]): planar (undirected) network single source and sink no vertex removals Introduction Current state of the art 8 / 29

19 s-t planar graphs Correspondence Elementary s-t cuts in G paths from s D to t D in G Value of cut equals dual length (λ ) of corresponding dual path. Introduction Current state of the art 9 / 29

20 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 ) = min (U \ X ) c (X ) B} X E {λ Theorem ν max B (G) = min{λ B (P ) P path from s D to t D } Introduction Current state of the art 10 / 29

21 Reduction to multi-objective shortest path problem (MOSP) ν max B (G) = min{λ (P ) P path from s D to t D in G, c (P ) B} Introduction Current state of the art 11 / 29

22 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

23 Characterizing countercl.w. s-t sep. circuits P: path from s to t in G P D = {e D e P} PR D = {ed R e P} Definition (Parity w.r.t. P) pp(u ) = U P D U PR D Theorem Let C be a circuit in G. C is counterclockwise s-t sep. p P (C ) = 1 Introduction Current state of the art 13 / 29

24 Characterizing countercl.w. s-t sep. circuits P: path from s to t in G P D = {e D e P} PR D = {ed R e P} Definition (Parity w.r.t. P) pp(u ) = U P D U PR D Theorem Let C be a circuit in G. C is counterclockwise s-t sep. p P (C ) = 1 Introduction Current state of the art 13 / 29

25 Characterizing countercl.w. s-t sep. circuits P: path from s to t in G P D = {e D e P} PR D = {ed R e P} Definition (Parity w.r.t. P) pp(u ) = U P D U PR D Theorem Let C be a circuit in G. C is counterclockwise s-t sep. p P (C ) = 1 Introduction Current state of the art 13 / 29

26 Transformation to MOSP problem ν max B (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 additional objective: parity. Again, the corresponding MOSP problem can be solved in pseudo-polynomial time by dynamic programming. Introduction Current state of the art 14 / 29

27 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 by doubling the vertices. Multiple sources and sinks can be reduced to a single source & sink by introduction of a supersource and supersink. However, these constructions destroy planarity. Introduction Current state of the art 15 / 29

28 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 by doubling the vertices. Multiple sources and sinks can be reduced to a single source & sink by introduction of a supersource and supersink. However, these constructions destroy planarity. Introduction Current state of the art 15 / 29

29 Outline 1 Introduction Definition and Motivation Complexity results Current state of the art 2 Extensions to planar network interdiction Vertex interdiction and vertex capacities Network flow security with multiple sources and sinks Final thoughts on complexity of planar network interdiction 3 Conclusions

30 Generalizing s-t cuts Extensions to planar network interdiction Vertex interdiction and vertex capacities 16 / 29

31 Generalizing s-t cuts Extensions to planar network interdiction Vertex interdiction and vertex capacities 16 / 29

32 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 in G \ Q. Furthermore, the reduced value of Q is defined by u B (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

33 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 in G \ Q. Furthermore, the reduced value of Q is defined by u B (Q) := min{u(q \ X ) X Q, c(x ) B} (convention: u(v) = v V ). ν max B (G) = min{u B (Q) Q s-t separating set in G} Extensions to planar network interdiction Vertex interdiction and vertex capacities 16 / 29

34 Adapting the dual network Extensions to planar network interdiction Vertex interdiction and vertex capacities 17 / 29

35 Adapting the dual network Extensions to planar network interdiction Vertex interdiction and vertex capacities 17 / 29

36 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

37 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

38 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

39 Correspondence between reduced values G = (Ṽ = V V, Ẽ = E Ê, λ, c, p P ) where λ, c and p P are extensions of λ, c and p P. Extensions to planar network interdiction Vertex interdiction and vertex capacities 18 / 29

40 Correspondence between reduced values (2) Relations between G and G i) Q s-t separating sets in G u max B (Q) λ B ( C (Q)). ii) C counterclockwise s-t separating circuit in G u max (G \ Q( C )) λ B ( C ) The problem can be solved as in the case without vertex interdiction by transformation to a MOSP. Extensions to planar network interdiction Vertex interdiction and vertex capacities 19 / 29

41 Vertex capacities Vertex capacities can easily be included into the model by a slight modification of the extended dual graph. Extensions to planar network interdiction Vertex interdiction and vertex capacities 20 / 29

42 Outline 1 Introduction Definition and Motivation Complexity results Current state of the art 2 Extensions to planar network interdiction Vertex interdiction and vertex capacities Network flow security with multiple sources and sinks Final thoughts on complexity of planar network interdiction 3 Conclusions

43 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 νb max(g) < νmax (G)} When dealing with unit interdiction cost, the network flow security problem corresponds to determining if a network is n k secure. Extensions to planar network interdiction Multiple sources and sinks 21 / 29

44 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 νb max(g) < νmax (G)} When dealing with unit interdiction cost, the network flow security problem corresponds to determining if a network is n k secure. Extensions to planar network interdiction Multiple sources and sinks 21 / 29

45 Relation with network interdiction The network flow security problem (NFSP) and single source & sink network flow interdiction problem (SSSNFIP) can easily be reduced to each other on general (not necessarily planar) graphs. NFSP SSSNFIP: SSSNFIP NFSP: Binary search over budget. 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

46 Relation with network interdiction The network flow security problem (NFSP) and single source & sink network flow interdiction problem (SSSNFIP) can easily be reduced to each other on general (not necessarily planar) graphs. NFSP SSSNFIP: SSSNFIP NFSP: Binary search over budget. 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

47 Pseudo-polynomial algorithm for planar NFSP 1 Transform the problem into a interdiction problem on flow circulations by sending flow from the sources to the sinks on artificial arcs. 2 Reformulate the problem on a dual network that allows to incorporate lower bounds on capacities and transform it to a MOSP. Extensions to planar network interdiction Multiple sources and sinks 23 / 29

48 1. Passage to interd. problem on circulations Extensions to planar network interdiction Multiple sources and sinks 24 / 29

49 1. Passage to interd. problem on circulations Extensions to planar network interdiction Multiple sources and sinks 24 / 29

50 1. Passage to interd. problem on circulations û and ĉ are extensions of u and c with ĉ(ê) = ê T. Extensions to planar network interdiction Multiple sources and sinks 24 / 29

51 1. Passage to interd. problem on circulations û and ĉ are extensions of u and c with ĉ(ê) = ê T. For every interdiction set R E we have: There is a saturating flow in G \ R There is a circulation in Ĝ \ R. Extensions to planar network interdiction Multiple sources and sinks 24 / 29

52 2a. Incorporating lower bounds into the dual Theorem ([Miller and Naor, 1995]) Ĝ admits a valid circulation. Ĝ contains no negative circuit. Extensions to planar network interdiction Multiple sources and sinks 25 / 29

53 2a. Incorporating lower bounds into the dual Theorem ([Miller and Naor, 1995]) Ĝ admits a valid circulation. Ĝ contains no negative circuit. Extensions to planar network interdiction Multiple sources and sinks 25 / 29

54 2b. Transformation to MOSP The theorem of Miller & Naor can easily be extended to include the possibility of interdiction. Theorem ν max B (G) < 0 circuit Ĉ in Ĝ such that λ B(Ĉ ) < 0 Finding a circuit with negative reduced length in Ĝ can be transformed into a MOSP similar to the previous problems. Extensions to planar network interdiction Multiple sources and sinks 26 / 29

55 Outline 1 Introduction Definition and Motivation Complexity results Current state of the art 2 Extensions to planar network interdiction Vertex interdiction and vertex capacities Network flow security with multiple sources and sinks Final thoughts on complexity of planar network interdiction 3 Conclusions

56 Complexity for NFIP with mult. sources/sinks Is network interdiction on planar graphs with multiple sources and sinks strongly NP-complete? Extensions to planar network interdiction Complexity revisited 27 / 29

57 Complexity for NFIP with mult. sources/sinks Is network interdiction on planar graphs with multiple sources and sinks strongly NP-complete? We do not know. Extensions to planar network interdiction Complexity revisited 27 / 29

58 Complexity for NFIP with mult. sources/sinks Is network interdiction on planar graphs with multiple sources and sinks strongly NP-complete? We do not know. But it is at least as difficult as finding dense subgraphs of planar graphs (whose complexity is also a long standing open problem). Extensions to planar network interdiction Complexity revisited 27 / 29

59 Reducing k-densest subgraph problem to NFIP k-densest subgraph problem on planar graphs: Input: Undirected planar graph G = (V, E), k N Output: max{#edges in G[V ] V V, V = k} max{#edges in G[V ] V V, V = k} = ν max (G ) ν max k (G ) Extensions to planar network interdiction Complexity revisited 28 / 29

60 Reducing k-densest subgraph problem to NFIP k-densest subgraph problem on planar graphs: Input: Undirected planar graph G = (V, E), k N Output: max{#edges in G[V ] V V, V = k} max{#edges in G[V ] V V, V = k} = ν max (G ) ν max k (G ) Extensions to planar network interdiction Complexity revisited 28 / 29

61 Conclusions Network interdiction is strongly NP-complete. Pseudo-polynomial algorithms were only available for (undirected) planar graphs with a single source & sink and without vertex interdiction. Pseudo-polynomial algorithms on directed planar graphs for the following extensions were presented: Vertex interdiction & vertex capacities Multiple sources and sinks in the context of network security. Hardness-result/algorithm is missing for network interdiction on planar graphs with multiple sources and sinks. The problem is at least as hard as the k-densest subgraph problem on planar graphs. Conclusions 29 / 29

62 References I N. Assimakopoulos. A network interdiction model for hospital infection control. Computers in biology and medicine, 17(6): , P. M. Ghare, D. C. Montgomery, and W. C. Turner. Optimal interdiction policy for a flow network. Naval Research Logistics Quarterly, 18:37 45, G. L. Miller and J. Naor. Flow in planar graphs with multiple sources and sinks. SIAM J. Comput., 24(5): , ISSN doi: Conclusions 30 / 29

63 References II C. A. Phillips. The network inhibition problem. In STOC 93: Proceedings of the twenty-fifth annual ACM symposium on Theory of computing, pages , New York, NY, USA, ACM Press. ISBN doi: J. Salmeron, K. Wood, and R. Baldick. Analysis of electric grid security under terrorist thread. IEEE Transaction on Power Systems, 19(2): , R. K. Wood. Deterministic network interdiction. Mathematical and Computer Modeling, 17(2):1 18, Conclusions 31 / 29