Robust Flows over Time: Models and Complexity Results Corinna - - PowerPoint PPT Presentation

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Robust Flows over Time: Models and Complexity Results Corinna - - PowerPoint PPT Presentation

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Robust Flows over Time: Models and Complexity Results Corinna Gottschalk 1 , Arie Koster 1 , Frauke Liers 2 , Britta Peis 1 , Daniel Schmand 1 , Andreas Wierz 1 1 RWTH Aachen University, 2 Universitt Erlangen-Nrnberg January 2016, Aussois

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Robust Flows over Time: Models and Complexity Results

Corinna Gottschalk1, Arie Koster1, Frauke Liers2, Britta Peis1, Daniel Schmand1, Andreas Wierz1

1 RWTH Aachen University, 2Universität Erlangen-Nürnberg

January 2016, Aussois

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SLIDE 2

Maximum Flow over Time.

  • Graph G = (V, E), capacities u : E → Z+, traveltimes τ : E → Z+,

time horizon T.

  • Maximize total flow shipped within the time horizon.

τ = 2 τ = 2 τ = 1 τ = 1 τ = 2 τ = 1 τ = 1 t = 1 T = 7 u ≡ 1 s d

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SLIDE 3

Maximum Flow over Time.

  • Graph G = (V, E), capacities u : E → Z+, traveltimes τ : E → Z+,

time horizon T.

  • Maximize total flow shipped within the time horizon.
  • Select paths and dispatch intervals.

τ = 2 τ = 2 τ = 1 τ = 1 τ = 2 τ = 1 τ = 1 T = 7 u ≡ 1 s d [ , 1 ) , f = 1 [1, T), f = 1 [0, T), f = 1

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SLIDE 4

Maximum Flow over Time.

  • Graph G = (V, E), capacities u : E → Z+, traveltimes τ : E → Z+,

time horizon T.

  • Maximize total flow shipped within the time horizon.
  • Select paths and dispatch intervals.

τ = 2 τ = 2 τ = 1 τ = 1 τ = 2 τ = 1 τ = 1 t = 2 T = 7 u ≡ 1 s d

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SLIDE 5

Maximum Flow over Time.

  • Graph G = (V, E), capacities u : E → Z+, traveltimes τ : E → Z+,

time horizon T.

  • Maximize total flow shipped within the time horizon.
  • Select paths and dispatch intervals.

τ = 2 τ = 2 τ = 1 τ = 1 τ = 2 τ = 1 τ = 1 t = 3 T = 7 u ≡ 1 s d

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SLIDE 6

Maximum Flow over Time.

  • Graph G = (V, E), capacities u : E → Z+, traveltimes τ : E → Z+,

time horizon T.

  • Maximize total flow shipped within the time horizon.
  • Select paths and dispatch intervals.

τ = 2 τ = 2 τ = 1 τ = 1 τ = 2 τ = 1 τ = 1 t = 4 T = 7 u ≡ 1 s d

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SLIDE 7

Maximum Flow over Time.

  • Graph G = (V, E), capacities u : E → Z+, traveltimes τ : E → Z+,

time horizon T.

  • Maximize total flow shipped within the time horizon.
  • Select paths and dispatch intervals.

τ = 2 τ = 2 τ = 1 τ = 1 τ = 2 τ = 1 τ = 1 t = 5 T = 7 u ≡ 1 s d

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SLIDE 8

Maximum Flow over Time.

  • Graph G = (V, E), capacities u : E → Z+, traveltimes τ : E → Z+,

time horizon T.

  • Maximize total flow shipped within the time horizon.
  • Select paths and dispatch intervals.

τ = 2 τ = 2 τ = 1 τ = 1 τ = 2 τ = 1 τ = 1 t = 6 T = 7 u ≡ 1 s d

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SLIDE 9

Maximum Flow over Time.

  • Graph G = (V, E), capacities u : E → Z+, traveltimes τ : E → Z+,

time horizon T.

  • Maximize total flow shipped within the time horizon.
  • Select paths and dispatch intervals.

τ = 2 τ = 2 τ = 1 τ = 1 τ = 2 τ = 1 τ = 1 t = 7 T = 7 u ≡ 1 s d

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SLIDE 10
  • Natural time-indexed LP formulation.
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SLIDE 11
  • Natural time-indexed LP formulation.
  • How to solve it?
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SLIDE 12
  • Natural time-indexed LP formulation.
  • How to solve it? Use temporally repeated flows.

max

x≥0

  • P∈P

(T − τ(P)) xP s.t.

  • P∈P:e∈P

xP ≤ ue ∀e ∈ E

  • Well known to be optimal (c.f. Ford & Fulkerson ’62).
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SLIDE 13

Robust Maximum Flow over Time.

τ = 2 τ = 2 τ = 1 τ = 1 τ = 2 τ = 1 τ = 1 T = 7 u ≡ 1 s d [ , 3 ) , f = 1 / 3 [0, 3), f = 2/3 [0, 2), f = 2/3

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SLIDE 14

Robust Maximum Flow over Time.

τ = 2 τ = 2 τ = 1 τ = 1 τ = 2 τ = 1 τ = 1 T = 7 u ≡ 1 s d [ , 3 ) , f = 1 / 3 [0, 3), f = 2/3 [0, 2), f = 2/3

  • What happens if some edges fail?
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SLIDE 15

Robust Maximum Flow over Time.

τ = 2 τ = 2 τ = 1 τ = 1 τ = 2 τ = 1 τ = 1 T = 7 u ≡ 1 s d [ , 3 ) , f = 1 / 3 [0, 3), f = 2/3 [0, 2), f = 2/3

  • What happens if some edges fail?
  • At most Γ ∈ Z+ edges fail simultaneously.

→ Scenarios S ≔ {X ⊆ E : |X| ≤ Γ}.

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SLIDE 16

Robust Maximum Flow over Time.

τ = 2 τ = 2 τ = 1 τ = 1 τ = 2 τ = 1 τ = 1 T = 7 u ≡ 1 s d [ , 3 ) , f = 1 / 3 [0, 3), f = 2/3 [0, 2), f = 2/3

  • What happens if some edges fail?
  • At most Γ ∈ Z+ edges fail simultaneously.

→ Scenarios S ≔ {X ⊆ E : |X| ≤ Γ}.

  • Goal: Maximize the total throughput in the worst case.
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SLIDE 17

Robust Maximum Flow over Time.

  • How to compute optimal robust flows?
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SLIDE 18

Robust Maximum Flow over Time.

  • How to compute optimal robust flows?
  • Tractability?
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SLIDE 19

Robust Maximum Flow over Time.

  • How to compute optimal robust flows?
  • Tractability?
  • Are optimal robust flows nominal optimal?
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SLIDE 20

Robust Maximum Flow over Time.

  • How to compute optimal robust flows?
  • Tractability?
  • Are optimal robust flows nominal optimal?
  • Are temporally repeated flows still optimal for the general

robust model?

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SLIDE 21

Are Temporally Repeated Flows still optimal?

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SLIDE 22

Are Temporally Repeated Flows still optimal? NO! s v1 v2 d τ = 0 τ = 1 τ = 2 τ = 0 τ = 2 τ = 1 τ = 0 T = 3, Γ = 2

Figure: Only solid edges are attackable. Unit capacity.

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SLIDE 23

Are Temporally Repeated Flows still optimal? NO! s v1 v2 d τ = 0 τ = 1 τ = 2 τ = 0 τ = 2 τ = 1 τ = 0 T = 3, Γ = 2

Figure: Only solid edges are attackable. Unit capacity.

  • Opt sends one unit into each path of length 2 simultaneously.
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SLIDE 24

Are Temporally Repeated Flows still optimal? NO! s v1 v2 d τ = 0 τ = 1 τ = 2 τ = 0 τ = 2 τ = 1 τ = 0 T = 3, Γ = 2

Figure: Only solid edges are attackable. Unit capacity.

  • Opt sends one unit into each path of length 2 simultaneously.
  • Temporally repeated flow sends only 1/3 into each path.

⇒ Gap of T.

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SLIDE 25

Are Temporally Repeated Flows still optimal? NO! s v1 v2 d τ = 0 τ = 1 τ = 2 τ = 0 τ = 2 τ = 1 τ = 0 T = 3, Γ = 2

Figure: Only solid edges are attackable. Unit capacity.

  • Opt sends one unit into each path of length 2 simultaneously.
  • Temporally repeated flow sends only 1/3 into each path.

⇒ Gap of T.

  • Are there natural upper bounds?
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SLIDE 26

Upper bound on the Optimality Gap.

  • Introduce interdictor choice into the models.

max

x≥0

  • P∈P

(T − τ(P)) xP − λ s.t.

  • P∈P:e∈P

xP ≤ ue ∀e ∈ E

  • P∩z∅

xP ≤ λ ∀z ∈ S (Similar for the general model).

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SLIDE 27

Upper bound on the Optimality Gap.

  • Introduce interdictor choice into the models.

max

x≥0

  • P∈P

(T − τ(P)) xP − λ s.t.

  • P∈P:e∈P

xP ≤ ue ∀e ∈ E

  • P∩z∅

xP ≤ λ ∀z ∈ S (Similar for the general model).

  • Upper bound on the gap between the models?
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SLIDE 28

Upper bound on the Optimality Gap.

  • Introduce interdictor choice into the models.

max

x≥0

  • P∈P

(T − τ(P)) xP − λ s.t.

  • P∈P:e∈P

xP ≤ ue ∀e ∈ E

  • P∩z∅

xP ≤ λ ∀z ∈ S (Similar for the general model).

  • Upper bound on the gap between the models?

Theorem: For a k-coverable instance of robust maximum flow over time, an optimal temporally repeated flow is an O(k · log T)-approximation.

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SLIDE 29

k-Coverability (example).

  • Consider e ∈ E and all paths P0, P1, . . . traversing e.

s d τ = 0 τ = 4 τ = 0 e τ = τ = τ = 0 τ = 0 τ = 4 1 2 3 4 5 6

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SLIDE 30

k-Coverability (example).

  • Consider e ∈ E and all paths P0, P1, . . . traversing e.
  • Pi may utilize e during time interval Ie,P ≔ [τ<e(Pi), T − τ≥e(Pi)].

s d τ = 0 τ = 4 τ = 0 e τ = τ = τ = 0 τ = 0 τ = 4 1 2 3 4 5 6

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SLIDE 31

k-Coverability (example).

  • Consider e ∈ E and all paths P0, P1, . . . traversing e.
  • Pi may utilize e during time interval Ie,P ≔ [τ<e(Pi), T − τ≥e(Pi)].

s d τ = 0 τ = 4 τ = 0 e τ = τ = τ = 0 τ = 0 τ = 4 1 2 3 4 5 6

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SLIDE 32

k-Coverability (example).

  • Consider e ∈ E and all paths P0, P1, . . . traversing e.
  • Pi may utilize e during time interval Ie,P ≔ [τ<e(Pi), T − τ≥e(Pi)].

s d τ = 0 τ = 4 τ = 0 e τ = τ = τ = 0 τ = 0 τ = 4 1 2 3 4 5 6

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SLIDE 33

k-Coverability (example).

  • Consider e ∈ E and all paths P0, P1, . . . traversing e.
  • Pi may utilize e during time interval Ie,P ≔ [τ<e(Pi), T − τ≥e(Pi)].
  • “e is k-coverable if all intervals can be observed from at most k

points”.

s d τ = 0 τ = 4 τ = 0 e τ = τ = τ = 0 τ = 0 τ = 4 1 2 3 4 5 6

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SLIDE 34

k-Coverability (example).

  • Consider e ∈ E and all paths P0, P1, . . . traversing e.
  • Pi may utilize e during time interval Ie,P ≔ [τ<e(Pi), T − τ≥e(Pi)].
  • e is k-coverable if the Ie,P-induced interval graph can be covered by

not more than k cliques.

s d τ = 0 τ = 4 τ = 0 e τ = τ = τ = 0 τ = 0 τ = 4 1 2 3 4 5 6

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SLIDE 35

k-Coverability (example).

  • Consider e ∈ E and all paths P0, P1, . . . traversing e.
  • Pi may utilize e during time interval Ie,P ≔ [τ<e(Pi), T − τ≥e(Pi)].
  • e is k-coverable if the Ie,P-induced interval graph can be covered by

not more than k cliques.

1 2 3 4 5 6

Examples:

  • Every instance with time horizon T is at most T-coverable.
  • Directed acyclic graphs with maxP τ(P) ≤ T are 1-coverable.
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SLIDE 36

Proof Strategy for the Theorem. Temporally Repeated Full Model (x∗, λ∗) (P) (D) (α∗, β∗) (x, λ) (P’) (D’) (α, β) Primal mapping val(x∗, λ∗) = val(x, λ) Strong Duality Weak duality Graphical interpretation of duals. val(α, β) ≤ O(k log T)val(α∗, β∗)

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SLIDE 37

Variants.

  • General, non-discrete model.
  • Edges don’t fail entirely but are delayed by ∆e ∈ Z+.
  • Some complexity results & tractable special cases.