Robust Network Design with Several Traffic Scenarios Models and - - PowerPoint PPT Presentation

Model Heuristic Polyhedral Structure B&C

Robust Network Design with Several Traffic Scenarios

Models and Algorithms

Eduardo ´ Alvarez-Miranda1 Valentina Cacchiani1 Tim Dorneth2 Michael J¨ unger2 Frauke Liers2 Andrea Lodi1 Tiziano Parriani1 Daniel R. Schmidt2

1DEIS, Universit`

a di Bologna

2Institut f¨

ur Informatik, Universit¨ at zu K¨

• ln

Financial support is acknowledged from the Ateneo Italo-Tedesco VIGONI programme and the DFG 1

Model Heuristic Polyhedral Structure B&C

Outline

1 A Robust Network Design Model 2 A Large Neighborhood Search Heuristic 3 The Polyhedral Structure 4 A Branch-and-Cut Algorithm

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Model Heuristic Polyhedral Structure B&C

About Robust Network Design

Given:

undirected graph G = (V , E) cost vector c : E → Z≥0 K integer balance vectors b1, . . . , bK : V → Z ( “scenarios” )

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Model Heuristic Polyhedral Structure B&C

About Robust Network Design

Given:

undirected graph G = (V , E) cost vector c : E → Z≥0 K integer balance vectors b1, . . . , bK : V → Z ( “scenarios” )

Task:

find integer capacities u : E → Z≥0 s.t. there is directed bi-flow f i w.r.t. u for all i = 1, . . . , K.

(1) f i

u,v + f i v,u ≤ uu,v

for all {u, v} ∈ E (2)

• u∈δ(v)

(f i

u,v − f i v,u) = bi v

for all v ∈ V

minimize cTu

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Model Heuristic Polyhedral Structure B&C

Flow based IP-formulation

min

• {u,v}∈E

cu,vuu,v f i

u,v + f i v,u ≤ uu,v

for all {u, v} ∈ E, i = 1, . . . , K

• u∈δ(v)

(f i

u,v − f i v,u) = bi v

for all v ∈ V , i = 1, . . . , K uu,v ∈ Z≥0 for all {u, v} ∈ E f i

u,v, f i v,u ∈ Z≥0

for all {u, v} ∈ E, i = 1, . . . , K

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Model Heuristic Polyhedral Structure B&C

Applications

streaming networks with changing customer demands planning of communication networks, public transport

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Model Heuristic Polyhedral Structure B&C

Applications

streaming networks with changing customer demands planning of communication networks, public transport

Related work

non-robust, multi-commodity setting

[Atamt¨ urk, 2001] [Koster, Orlowski, Raack, Wess¨ aly 2007, 2008] [Avella, Mattia, Sassano 2007]

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Model Heuristic Polyhedral Structure B&C

Applications

streaming networks with changing customer demands planning of communication networks, public transport

Related work

non-robust, multi-commodity setting

[Atamt¨ urk, 2001] [Koster, Orlowski, Raack, Wess¨ aly 2007, 2008] [Avella, Mattia, Sassano 2007]

robust, multi-commodity setting

[Koster, Kutschka, Raack 2010, 2011]

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Model Heuristic Polyhedral Structure B&C

Applications

streaming networks with changing customer demands planning of communication networks, public transport

Related work

non-robust, multi-commodity setting

[Atamt¨ urk, 2001] [Koster, Orlowski, Raack, Wess¨ aly 2007, 2008] [Avella, Mattia, Sassano 2007]

robust, multi-commodity setting

[Koster, Kutschka, Raack 2010, 2011]

robust, single commodity, scenarios given by polytope

[Ben-Ameur, Kerivin 2005]

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Model Heuristic Polyhedral Structure B&C

Applications

streaming networks with changing customer demands planning of communication networks, public transport

Related work

non-robust, multi-commodity setting

[Atamt¨ urk, 2001] [Koster, Orlowski, Raack, Wess¨ aly 2007, 2008] [Avella, Mattia, Sassano 2007]

robust, multi-commodity setting

[Koster, Kutschka, Raack 2010, 2011]

robust, single commodity, scenarios given by polytope

[Ben-Ameur, Kerivin 2005]

robust, single commodity

[Buchheim, Liers, Sanit` a 2008] [Oriolo, Sanit` a]

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Model Heuristic Polyhedral Structure B&C

Complexity

polynomial cases

single scenario, multiple sources/sinks (minimum cost flow)

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Model Heuristic Polyhedral Structure B&C

Complexity

polynomial cases

single scenario, multiple sources/sinks (minimum cost flow)

Problem is already NP-hard if. . .

K = 3 scenarios, binary demands, single-source [Sanit` a 2009] K = 2 scenarios, multiple sources/sinks [Sanit` a 2009]

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Model Heuristic Polyhedral Structure B&C

Complexity

polynomial cases

single scenario, multiple sources/sinks (minimum cost flow)

Problem is already NP-hard if. . .

K = 3 scenarios, binary demands, single-source [Sanit` a 2009] K = 2 scenarios, multiple sources/sinks [Sanit` a 2009]

• pen cases

K = 2 scenarios, binary demands, single-source

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Model Heuristic Polyhedral Structure B&C

Example (unit edge costs)

Scenario 1 2

• 2

c ≡ 1

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Model Heuristic Polyhedral Structure B&C

Example (unit edge costs)

Scenario 1 2

• 2

c ≡ 1 Scenario 2 1

• 1

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Model Heuristic Polyhedral Structure B&C

Example (unit edge costs)

Scenario 1 2

• 2

2

c ≡ 1 Scenario 2 1

• 1

1 1

Feasible solution:

2 1 1

cost: 4

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Model Heuristic Polyhedral Structure B&C

Example (unit edge costs)

Scenario 1 2

• 2

2

c ≡ 1 Scenario 2 1

• 1

1 1

Feasible solution:

2 1

cost: 3

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Model Heuristic Polyhedral Structure B&C

Heuristic: Constructive Phase

1 insert auxilliary edge (for all edges e ∈ E)

ce

• ce/∞

0/0

capacity that is already installed additional capacity must be paid

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Model Heuristic Polyhedral Structure B&C

Heuristic: Constructive Phase

1 insert auxilliary edge (for all edges e ∈ E)

ce

• ce/∞

0/0

capacity that is already installed additional capacity must be paid 2 For q = 1, . . . , K: 1 compute MinCost flow for q-th scenario

fe

2 update capacities for subsequent scenarios

ce/∞ 0/(ue + fe)

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Model Heuristic Polyhedral Structure B&C

Heuristic: Constructive Phase

1 insert auxilliary edge (for all edges e ∈ E)

ce

• ce/∞

0/0

capacity that is already installed additional capacity must be paid 2 For q = 1, . . . , K: Caution: The order matters! 1 compute MinCost flow for q-th scenario

fe

2 update capacities for subsequent scenarios

ce/∞ 0/(ue + fe)

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Model Heuristic Polyhedral Structure B&C

Heuristic: Improvement Phase

Large Neighborhood Search

use constructive phase information to constrain variables in IP find good solution“close”to constructed solution control closeness (neighborhood size) by parameter T

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Model Heuristic Polyhedral Structure B&C

Heuristic: Improvement Phase

Large Neighborhood Search

use constructive phase information to constrain variables in IP find good solution“close”to constructed solution control closeness (neighborhood size) by parameter T

Definition

Let f 1, . . . , f K be flows from constructive phase Set Ue := max{f i

e | i = 1, . . . , K} for all e ∈ E

Set E ′ := E \ {e ∈ E | f i

e = 0 ∀i}

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Model Heuristic Polyhedral Structure B&C

Heuristic: Improvement Phase

Large Neighborhood Search

use constructive phase information to constrain variables in IP find good solution“close”to constructed solution control closeness (neighborhood size) by parameter T

Definition

Let f 1, . . . , f K be flows from constructive phase Set Ue := max{f i

e | i = 1, . . . , K} for all e ∈ E

Set E ′ := E \ {e ∈ E | f i

e = 0 ∀i}

Algorithm

Limit ue by Ue plus some limited tolerance (neighborhood) Solve flow-based IP on (V , E ′).

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Model Heuristic Polyhedral Structure B&C

Improvement phase IP

min

• {u,v}∈E ′

cu,v(Uu,v + wu,v)

• u∈δ(v)

(f i

u,v − f i v,u) = bi v

for all v ∈ V , i = 1, . . . , K f i

u,v + f i v,u ≤ Uu,v + wu,v

for all {u, v} ∈ E ′, i = 1, . . . , K

• {u,v}∈E ′

wu,v ≤ T wu,v ≥ 0 for all {u, v} ∈ E ′ uu,v ∈ Z≥0 for all {u, v} ∈ E ′ f i

u,v, f i v,u ∈ Z≥0

for all {u, v} ∈ E ′, i = 1, . . . , K

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Model Heuristic Polyhedral Structure B&C

Difficult instances

Instances on d-dimensional hypercube

• pposite nodes on hypercube are terminal pairs

set random supply from 1, . . . , 10

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Model Heuristic Polyhedral Structure B&C

Difficult instances

Instances on d-dimensional hypercube

• pposite nodes on hypercube are terminal pairs

2

• 2

set random supply from 1, . . . , 10

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Model Heuristic Polyhedral Structure B&C

Difficult instances

Instances on d-dimensional hypercube

• pposite nodes on hypercube are terminal pairs

5

• 5

set random supply from 1, . . . , 10

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Model Heuristic Polyhedral Structure B&C

Difficult instances

Instances on d-dimensional hypercube

• pposite nodes on hypercube are terminal pairs

1

• 1

set random supply from 1, . . . , 10

11

Model Heuristic Polyhedral Structure B&C

Difficult instances

Instances on d-dimensional hypercube

• pposite nodes on hypercube are terminal pairs

8

• 8

set random supply from 1, . . . , 10

11

Model Heuristic Polyhedral Structure B&C

Difficult instances

Instances on d-dimensional hypercube

• pposite nodes on hypercube are terminal pairs

8

• 8

set random supply from 1, . . . , 10 if all supplies 1: integrality gap converges to 2 as d → ∞

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Model Heuristic Polyhedral Structure B&C

Computational Results

Cplex 12.3, CS2 code by Goldberg for CP Intel(R) Core(TM) i7 CPU, 64 bit, 1.73 GHz, 6 Gb RAM time limit of 7200 sec. average values

12

Model Heuristic Polyhedral Structure B&C

Computational Results

Cplex 12.3, CS2 code by Goldberg for CP Intel(R) Core(TM) i7 CPU, 64 bit, 1.73 GHz, 6 Gb RAM time limit of 7200 sec. average values

CP RNDinit T = ∞ T = 25 dim. time %gap time %gap time %gap time %gap 6d 0.06s 16.10 1533.59s 0.25 21.82s 1.02 16.29 1.02 7d 0.24s 25.78 7200.00s 10.70 3003.73s 2.13 448.77s 4.02

if all demands are 1: preprocessing necessary

12

Model Heuristic Polyhedral Structure B&C

Outline

1 A Robust Network Design Model 2 A Large Neighborhood Search Heuristic 3 The Polyhedral Structure 4 A Branch-and-Cut Algorithm

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Model Heuristic Polyhedral Structure B&C

Cut-Set inequalities for RND

Definition

Let S ⊆ V . Then we call

• {u,v}∈δ(S)

uu,v ≥ max

q=1,...,K |

• v∈S

bq

v|

the cut-set-inequality of S for some capacity vector u.

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Model Heuristic Polyhedral Structure B&C

Cut-Set inequalities for RND

Definition

Let S ⊆ V . Then we call

• {u,v}∈δ(S)

uu,v ≥ max

q=1,...,K |

• v∈S

bq

v|

the cut-set-inequality of S for some capacity vector u. 1 3 3 x

• 2
• 3
• 2

S

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Model Heuristic Polyhedral Structure B&C

Cut-Set inequalities for RND

Definition

Let S ⊆ V . Then we call

• {u,v}∈δ(S)

uu,v ≥ max

q=1,...,K |

• v∈S

bq

v|

the cut-set-inequality of S for some capacity vector u. 1 3 3 x

• 2
• 3
• 2

u1 u2 S

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Model Heuristic Polyhedral Structure B&C

Cut-Set inequalities for RND

Definition

Let S ⊆ V . Then we call

• {u,v}∈δ(S)

uu,v ≥ max

q=1,...,K |

• v∈S

bq

v|

the cut-set-inequality of S for some capacity vector u. 1 3 3 x

• 2
• 3
• 2

u1 u2 S u1 + u2 ≥ 3

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Model Heuristic Polyhedral Structure B&C

Cut-Set-Formulation

Theorem

Every instance (c; b1, . . . , bK) of the RND problem can be written as the following integer linear program: min

• {u,v}∈E

cu,vuu,v

• {u,v}∈δ(S)

uu,v ≥ max

q=1,...,K |

• v∈S

bq

v|

for all S ⊆ V uu,v ∈ Z≥0 for all {u, v} ∈ E

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Model Heuristic Polyhedral Structure B&C

A helpful Fact

S+ R

Let

b be balance vector,

v∈V bv = 0.

S+ := {v ∈ V | bv > 0} f flow R :=“all nodes reachable from S+ in residual network”

Then

If some residual demand left. . . can modify f such that. . . strictly less than |

v∈R bv| units of flow are sent out of R.

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Model Heuristic Polyhedral Structure B&C

Cut-Set-Formulation: Proof.

The cut-set inequalities are sufficient

let u satisfy the cut-set inequalities for all S ⊆ V assume that not all demands can be met in some scenario k run maximum flow algorithm for scenario k

• btain flow f and residual network Nf

set S+ and R as before. but then:

• e∈δ(R)

ue =

• {u,v}∈δ(R)

(fu,v + fv,u) =

• {u,v}∈δ(R)

u∈R

fu,v

(Fact)

<

• u∈R bk

u

• 17

Model Heuristic Polyhedral Structure B&C

Separation of Cut-Set-Inequalities

b1 b2 b3

Cutting off u∗

1 separate for each scenario k

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Model Heuristic Polyhedral Structure B&C

Separation of Cut-Set-Inequalities

b1 b2 b3

Cutting off u∗

1 separate for each scenario k

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Model Heuristic Polyhedral Structure B&C

Separation of Cut-Set-Inequalities

b1 b2 b3

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

Cutting off u∗

1 separate for each scenario k 2 for e ∈ E set w(e) = u∗ e

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Model Heuristic Polyhedral Structure B&C

Separation of Cut-Set-Inequalities

s b1 b2 b3

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

Cutting off u∗

1 separate for each scenario k 2 for e ∈ E set w(e) = u∗ e 3 insert node s

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Model Heuristic Polyhedral Structure B&C

Separation of Cut-Set-Inequalities

s b1 b2 b3

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

Cutting off u∗

1 separate for each scenario k 2 for e ∈ E set w(e) = u∗ e 3 insert node s 4 insert edge {s, τ} for all terminals τ

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Model Heuristic Polyhedral Structure B&C

Separation of Cut-Set-Inequalities

s b1 b2 b3

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

− b1 − b2 − b

3

Cutting off u∗

1 separate for each scenario k 2 for e ∈ E set w(e) = u∗ e 3 insert node s 4 insert edge {s, τ} for all terminals τ 5 set w(s, τ) = −bk τ (caution!)

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Model Heuristic Polyhedral Structure B&C

Separation of Cut-Set-Inequalities

s b1 b2 b3

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

− b1 − b2 − b

3

Cutting off u∗

1 separate for each scenario k 2 for e ∈ E set w(e) = u∗ e 3 insert node s 4 insert edge {s, τ} for all terminals τ 5 set w(s, τ) = −bk τ (caution!) 6 compute Min-s-Cut X w.r.t. w

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Model Heuristic Polyhedral Structure B&C

Separation of Cut-Set-Inequalities

s b1 b2 b3

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

− b1 − b2 − b

3

Cutting off u∗

1 separate for each scenario k 2 for e ∈ E set w(e) = u∗ e 3 insert node s 4 insert edge {s, τ} for all terminals τ 5 set w(s, τ) = −bk τ (caution!) 6 compute Min-s-Cut X w.r.t. w 7 if val(X) < 0, found inequality with

• max. violation

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Model Heuristic Polyhedral Structure B&C

Separation of Cut-Set-Inequalities

s b1 b2 b3

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

u∗

e

− b1 − b2 − b

3

Cutting off u∗

1 separate for each scenario k 2 for e ∈ E set w(e) = u∗ e 3 insert node s 4 insert edge {s, τ} for all terminals τ 5 set w(s, τ) = −bk τ (caution!) 6 compute Min-s-Cut X w.r.t. w 7 if val(X) < 0, found inequality with

• max. violation

Good news

Minimum Cut in star negative graph is polynomial [McCormick, Rao, Rinaldi 2003]

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Model Heuristic Polyhedral Structure B&C

Two IP formulations

flow formulation

linearization of non-linear problem difficult to gain polyhedral knowledge but: polynomial size, intuitiv can be seen as an extended formulation

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Model Heuristic Polyhedral Structure B&C

Two IP formulations

flow formulation

linearization of non-linear problem difficult to gain polyhedral knowledge but: polynomial size, intuitiv can be seen as an extended formulation

cut-set formulation

less variables in practice: description easier to understand exponential size, but separation in polynomial time

19

Model Heuristic Polyhedral Structure B&C

Results: Branch-and-Cut

Intel(R) Core2Duo(TM) E7500 @2.93Ghz CPU, 64bit 4GB RAM time limit of 600s CPLEX 12.1, ABACUS 3.2 (beta)

Results

2d, 3d, 4d: Solved to optimality in < 1s 5d: 4 out of 10 solved to optimality in < 70s 6d: no bound within time limit

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Model Heuristic Polyhedral Structure B&C

Outlook & Conclusion

What we have seen

versatile heuristic, can trade running time for quality class of difficult instances with provable integrality gap new IP formulation that leads to branch-and-cut algorithm there is a new version of ABACUS!

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Model Heuristic Polyhedral Structure B&C

Outlook & Conclusion

What we have seen

versatile heuristic, can trade running time for quality class of difficult instances with provable integrality gap new IP formulation that leads to branch-and-cut algorithm there is a new version of ABACUS!

What still needs work

tests on more instances more insights about polyhedral structure incorporate heuristic into B&C (with tuning)

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Model Heuristic Polyhedral Structure B&C

The end.

22