Protection of flows Protection of flows under targeted attacks - - PowerPoint PPT Presentation

Protection of flows under targeted attacks Protection of flows under targeted attacks

Jannik Matuschke TU Berlin Tom McCormick UBC Vancouver Gianpaolo Oriolo Tor Vergata Britta Peis RWTH Aachen Martin Skutella TU Berlin

Robust flows

protect flow against link failures/targeted attacks

Robust flows

protect flow against link failures/targeted attacks Adversarial model (1) flow player: specify nominal flow max (2) interdictor: reduce flow min

Robust flows

protect flow against link failures/targeted attacks Adversarial model (1) flow player: specify nominal flow max (2) interdictor: reduce flow min This talk: new model with fractional flow interdiction

Outline 1 Fractional interdiction model 2 Flow fortification

The model s

2 3 2 t 1 3 u ≡ 2

capacitated network with interdiction costs

◮ c(e): cost of removing 1 unit flow from e

The model s

2 3 2 t 1 3 u ≡ 2 B = 2

capacitated network with interdiction costs

◮ c(e): cost of removing 1 unit flow from e ◮ B : interdictor budget

The model s

2 3 2 t 1 3 u ≡ 2 B = 2

(1) flow player: specify path flow x(P)

The model s

2 3 2 t 1 3 u ≡ 2 B = 2

(1) flow player: specify path flow x(P) (2) interdictor: specify removed flow z(P)

The model s

2 3 2 t 1 3 u ≡ 2 B = 2

(1) flow player: specify path flow x(P) (2) interdictor: specify removed flow z(P) interdictor greedily attacks weakest link: ¯ c(P) := mine∈P c(e)

The model s

2 3 2 t 1 3 u ≡ 2 B = 2

(1) flow player: specify path flow x(P) (2) interdictor: specify removed flow z(P) interdictor greedily attacks weakest link: ¯ c(P) := mine∈P c(e)

The model s

2 3 2 t 1 3 u ≡ 2 B = 2

(1) flow player: specify path flow x(P) (2) interdictor: specify removed flow z(P) interdictor greedily attacks weakest link: ¯ c(P) := mine∈P c(e) Z := {z : 0 ≤ z(P) ≤ x(P),

P ¯

c(P)z(P) ≤ B} robust value: valr(x) := minz∈Z

• P x(P) − z(P)

The model s

2 3 2 t 1 3 u ≡ 2 B = 2 z(P1) = 1

(1) flow player: specify path flow x(P) (2) interdictor: specify removed flow z(P) interdictor greedily attacks weakest link: ¯ c(P) := mine∈P c(e) Z := {z : 0 ≤ z(P) ≤ x(P),

P ¯

c(P)z(P) ≤ B} robust value: valr(x) := minz∈Z

• P x(P) − z(P)

The model s

2 3 2 t 1 3 u ≡ 2 B = 2 z(P1) = 1 z(P2) = 1/2

(1) flow player: specify path flow x(P) (2) interdictor: specify removed flow z(P) interdictor greedily attacks weakest link: ¯ c(P) := mine∈P c(e) Z := {z : 0 ≤ z(P) ≤ x(P),

P ¯

c(P)z(P) ≤ B} robust value: valr(x) := minz∈Z

• P x(P) − z(P)

The model s

2 3 2 t 1 3 u ≡ 2 B = 2 z(P1) = 1 z(P2) = 1/2 valr(x) = 3/2

(1) flow player: specify path flow x(P) (2) interdictor: specify removed flow z(P) interdictor greedily attacks weakest link: ¯ c(P) := mine∈P c(e) Z := {z : 0 ≤ z(P) ≤ x(P),

P ¯

c(P)z(P) ≤ B} robust value: valr(x) := minz∈Z

• P x(P) − z(P)

Optimal flow strategy

Algorithm

◮ guess C∗ := max{¯

c(P) : z∗(P) > 0} for optimal (x∗, z∗)

Optimal flow strategy

Algorithm

◮ guess C∗ := max{¯

c(P) : z∗(P) > 0} for optimal (x∗, z∗)

◮ c′(P) := min{¯

c(P), C∗}

Optimal flow strategy

Algorithm

◮ guess C∗ := max{¯

c(P) : z∗(P) > 0} for optimal (x∗, z∗)

◮ c′(P) := min{¯

c(P), C∗}

◮ find flow x′ maximizing P c′(P)x′(P)

Optimal flow strategy

Algorithm

◮ guess C∗ := max{¯

c(P) : z∗(P) > 0} for optimal (x∗, z∗)

◮ c′(P) := min{¯

c(P), C∗}

◮ find flow x′ maximizing P c′(P)x′(P)

Theorem

x′ is an optimal robust flow.

Optimal flow strategy

Algorithm

◮ guess C∗ := max{¯

c(P) : z∗(P) > 0} for optimal (x∗, z∗)

◮ c′(P) := min{¯

c(P), C∗}

◮ find flow x′ maximizing P c′(P)x′(P)

Theorem

x′ is an optimal robust flow. Proof idea: valr(x∗) =

• P

x∗(P) − z∗(P)

Optimal flow strategy

Algorithm

◮ guess C∗ := max{¯

c(P) : z∗(P) > 0} for optimal (x∗, z∗)

◮ c′(P) := min{¯

c(P), C∗}

◮ find flow x′ maximizing P c′(P)x′(P)

Theorem

x′ is an optimal robust flow. Proof idea: valr(x∗) =

• P

x∗(P) − z∗(P) =

• P

c′(P) C∗ (x∗(P) − z∗(P))

Optimal flow strategy

Algorithm

◮ guess C∗ := max{¯

c(P) : z∗(P) > 0} for optimal (x∗, z∗)

◮ c′(P) := min{¯

c(P), C∗}

◮ find flow x′ maximizing P c′(P)x′(P)

Theorem

x′ is an optimal robust flow. Proof idea: valr(x∗) =

• P

x∗(P) − z∗(P) =

• P

c′(P) C∗ (x∗(P) − z∗(P)) =

• P c′(P)x∗(P)

C∗ − B C∗

Optimal flow strategy

Algorithm

◮ guess C∗ := max{¯

c(P) : z∗(P) > 0} for optimal (x∗, z∗)

◮ c′(P) := min{¯

c(P), C∗}

◮ find flow x′ maximizing P c′(P)x′(P)

Theorem

x′ is an optimal robust flow. Proof idea: valr(x∗) =

• P

x∗(P) − z∗(P) =

• P

c′(P) C∗ (x∗(P) − z∗(P)) =

• P c′(P)x∗(P)

C∗ − B C∗ ≤ valr(x′)

Further results

G

∞

s s1

1

s2

2

t1 t2 t

1 2

◮ reduction from multi-commodity flow

◮ integral version NP-hard ◮ preserves combinatorial algorithms

◮ generalizes to packing problems on arbitrary set systems

◮ multi-commodity flow ◮ abstract flows ◮ b-matchings

Flow fortification

(1) flow player: specify path flow x(P) for P ∈ P and added interdiction costs c+(e) for e ∈ E

◮ fortification cost γ(e), fortification budget BF ◮ e∈E γ(e)c+(e)x(e) ≤ BF

(2) interdictor: attack flow on paths z(P) for P ∈ P

◮ cost of interdicting flow unit on P:

¯ c(P) := min

e∈P c0(e) + c+(e) ◮ fractionally interdict as before

s 2+1 3+0 2+1 t 1+1 3+0

Results for fortification variant

Theorem

It is NP-hard to decide whether OPT > 0. Proof idea: reduction from Disjoint Paths

Theorem

If c0 ≡ 0, then optimal flow and fortification strategy can be found in polynomial time. Proof idea:

◮ optimal fortification strategy: uniform interdiction cost

C+(x) := BF

• e γ(e)x(e)

◮ optimal flow: maximizes P x(P) − B/C+(x)

(can be computed by min cost circulation)

Summary goal: robustness against targeted attacks

New: model with fractional interdiction

◮ max robust flow in poly-time ◮ fortification: hard in general, but tractable in special case

Summary goal: robustness against targeted attacks

New: model with fractional interdiction

◮ max robust flow in poly-time ◮ fortification: hard in general, but tractable in special case

Thank you!