Network Models for Disaster Managementplaza.ufl.edu/ashwin/slides.pdf · Network Models for Disaster Management Ashwin Arulselvan Industrial and Systems Engineering, Candidacy Admission

IntroductionCritical Node Detection

Review of Evacuation ProblemsNetwork Flows with Lane Reversals

Multimodal Evacuation Problem

Network Models for Disaster Management

Ashwin Arulselvan

Industrial and Systems Engineering, Candidacy Admission

June 5, 2008

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Outline

1 Introduction

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Outline

1 Introduction

2 Critical Node DetectionIntroductionCritical Node DetectionCardinality Constrained Node Detection ProblemSummary

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Outline

1 Introduction


3 Review of Evacuation Problems

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Outline

1 Introduction



4 Network Flows with Lane ReversalsSingle Source and Single SinkMultiple Sources and Multiple SinksSummary

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Outline

1 Introduction




5 Multimodal Evacuation ProblemIntroductionFormulationBranch and Price

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Outline

1 Introduction





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Introduction

• Need for disaster studies

• Proactive and Reactive Strategies

• Critical Node Detection Problem

• Network flows with lane reversals

• Multimodal Evacuation

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Contributions

A. Arulselvan, C.W. Commander, L. Elefteriadou, P.M. Pardalos.Detecting critical nodes in social networks.Computers and Operations Research, 2008.

A. Arulselvan, C.W. Commander, P.M. Pardalos, O. Shylo.Managing network risk via critical node identification.Risk Management in Telecommunication Networks, N. Gulpinar and B. Rustem(editors), Springer, to appear 2008 (in process)

A. Arulselvan, L. Elefteriadou, P. Pardalos.Review of Evacuation Problems.Submitted to Transport Reviews, 2008.

S. Rebennack, A. Arulselvan, L. Elefteriadou, P. M. Pardalos.Complexity Analysis for Maximum Flow Problems with Arc Reversals.Submitted to Networks, 2008.

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IntroductionCritical Node DetectionCardinality Constrained Node Detection ProblemSummary

Outline

1 Introduction





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Introduction

• Critical Node Detection

• Centrality

• Prestige

• Prominence

• Key Players

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Applications

• Assessing vulnerability of a supply chain network bydetermining the vital nodes

• First of many problems considered involvingjamming/suppressing communication on a network

• Breakdown communication in covert networks

• Reduce transmissibility of virus and contagion of epidemic

• Drug design

• Emergency response

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Problem Definition

• Decision Version: K − CNP

• Input: Undirected graph G = (V ,E ) and integers k and K

• Question: Is there a set M, where M is the set of all maximalconnected components of G obtained by deleting k nodes orless, such that ∑

∀i∈M

σi (σi − 1)

2≤ K

where σi is the cardinality of component i, for all i in M?

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NP-Completeness

LemmaLet M be a partition of G = (V ,E ) in to L components obtainedby deleting a set D, where |D| = k. Then the objective function

∑

∀i∈M

σi (σi − 1)

2≥

(|V | − k)( (|V |−k)L − 1)

2

with equality holding if and only σi = σj , for all i , j ∈ M, where σi

is the size of i th component of M.

• Objective function is best when components are of averagesize.

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NP-Completeness

LemmaLet M1 and M2 be a two sets of partitions of G = (V,E) obtainedby deleting a set D1 and D2 sets of nodes respectively, where|D1| = |D2| = k. Let L1 and L2 be the number of components inM1 and M2 respectively, and L1 ≥ L2. If σi = σj ,∀i , j ∈ M1, thenwe obtain a better objective function value by deleting D1.

• THE MORE (components), THE BETTER!!

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NP-Completeness

TheoremThe K − CNP problem is NP-complete.

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Formulation

• Let u i ,j = 1, if i and j are in the same component of G (V A),and 0 otherwise.

• Let v i = 1, if node i is deleted in the optimal solution, and 0otherwise.

• We can formulate the CNP as the following integer linearprogram

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Formulation

Minimize∑

i,j∈V

uij (1)

s.t.

uij + vi + vj ≥ 1, ∀ (i , j) ∈ E , (2)

uij + ujk − uki ≤ 1, ∀ (i , j , k) ∈ V , (3)

uij − ujk + uki ≤ 1, ∀ (i , j , k) ∈ V , (4)

−uij + ujk + uki ≤ 1, ∀ (i , j , k) ∈ V , (5)∑

i∈V

vi ≤ k, (6)

uij ∈ {0, 1}, ∀ i , j ∈ V , (7)

vi ∈ {0, 1}, ∀ i ∈ V . (8)

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Heuristics

• Notice if objective was only a function of the number ofcomponents then we could use approximation for Max K-Cutby modifying the Gomory-Hu tree

• BUT objective is also concerned with size of components

• Problem is harder

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Heuristics

• Recall the objective function:• Minimize

∑j∈V ,j 6=i

∑i∈V uij , where uij = 1, if i and j in same

component of vertex deleted subgraph.

• We can re-write this as follows:

• Where S is the set of all components and si is the size of thei th component.

• Easily identify with DFS in O(|V | + |E |) time!

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Heuristics

• We implement a heuristic based on Maximal Independent Sets

• Why? Because induced subgraph is empty• Maximum Independent Set provides upper bound on # of

components in optimal solution.

• Greedy type procedure

• Enhanced with local search procedure

• Results• Heuristic obtains optimal solutions in less time compared to

CPLEX• Runs in O(k2 + |V |k) time.

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Heuristics

procedure CriticalNode(G , k) MIS← MaximalIndepSet(G) while (|MIS| 6= |V | − k) do

i ← arg min{∑

j∈S

σj (σj−1)

2: S ∈ G(MIS ∪ {j}), j ∈ V \MIS

}

MIS← MIS ∪ {i} end while

return V \MIS /∗ set of k nodes to delete ∗/end procedure CriticalNode

Figure: Heuristic for detecting critical nodes.

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Heuristics

procedure LocalSearch(V \MIS) X∗ ← MIS

local improvement ← .TRUE. while local improvement do

local improvement ← .FALSE. if i ∈ MIS and j 6∈ MIS then

MIS← MIS \ i MIS← MIS ∪ j

if f (MIS) < f (X∗) then

X∗ ← MIS

local improvement ← .TRUE. else

MIS← MIS \ j /∗ undo swap ∗/ MIS← MIS ∪ i end if

end if

end while

return (V \ X∗) /∗ set of k nodes to delete ∗/end procedure LocalSearch

Figure: Local search algorithm for critical node heuristic.

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Results - Terrorist Network Data from Krebs

Figure: Terrorist network compiled by Krebs.

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Figure: Optimal solution when k = 20.

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Instance IP Model HeuristicNodes Objective Execution Objective Execution

Deleted (k) Value Time (s) Value Time (s)20 20 12.69 20 0.0115 61 277.77 61 0.0110 169 3337.06 169 0.029 214 2792.33 214 0.028 282 15111.94 282 0.017 327 10792.08 327 0.01

Optimal Solution found for all instances

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Results - Randomly Generated Scale Free Graphs

Instance IP Model Heuristic Heuristic + LSNodes Arcs Deleted Obj Comp Obj Comp Obj Comp

Nodes (k) Value Time (s) Value Time (s) Value Time (s)75 140 20 36 66.7 92 0.12 36 0.0375 140 25 18 33.28 39 0.28 18 0.0375 140 30 7 4.23 18 0.02 7 0.0475 210 25 26 93.71 78 0.1 26 0.0475 210 30 8 3.57 31 0.05 8 0.0575 210 35 2 4.36 16 0.18 2 0.0475 280 33 26 749.19 54 0.00 26 0.0475 280 35 20 164.34 38 0.09 20 0.0675 280 37 13 83.98 24 0.39 13 0.11100 194 25 44 151.14 142 0.731 44 0.09100 194 30 20 59.66 72 0.56 20 0.11100 194 35 10 8.51 33 0.66 10 0.12100 285 40 23 136.47 48 1.151 23 0.11100 285 42 17 263.82 38 0.4 17 0.17100 285 45 11 16.78 29 0.53 11 0.23

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Results - Randomly Generated Scale Free Graphs

Instance IP Model Heuristic Heuristic + LSNodes Arcs Deleted Obj Comp Obj Comp Obj Comp

Nodes (k) Value Time (s) Value Time (s) Value Time (s)100 380 45 22 128.13 58 0.58 22 0.15100 380 47 16 243.07 42 1.191 16 0.16100 380 50 10 228.72 23 0.31 10 0.11125 240 33 62 5047.51 97 0.721 62 0.30125 240 40 29 118.92 49 1.562 29 0.24125 240 45 16 17.09 32 0.14 16 0.39150 290 40 40 41.6 125 1.832 40 0.47150 290 50 12 26.29 64 2.773 12 0.831150 290 60 1 24.92 35 1.091 1 0.851150 435 61 19 29.55 53 2.313 19 0.741150 435 65 13 31.45 37 0.991 13 1.952150 435 67 11 37.91 31 0.52 11 0.801

• Optimal solution found for each instance

• Average CPLEX time: 289.44 seconds

• Average Heuristic time: 0.33 seconds

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Cardinality Critical Node Problem (CCNP)

• Alternate Formulation:• Suppose now, we want to limit the connectivity of the agents.• We can impose a constraint for this.• Now, we minimize the number of nodes deleted to satisfy this

constraint.• We have the CARDINALITY CONSTRAINED CRITICAL

NODE PROBLEM• The problem is still NP-hard

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CCNP - Formulation

Minimize∑

i∈V

vi (9)

s.t.

uij + vi + vj ≥ 1, ∀ (i , j) ∈ E , (10)

uij + ujk − uki ≤ 1, ∀ (i , j , k) ∈ V , (11)

uij − ujk + uki ≤ 1, ∀ (i , j , k) ∈ V , (12)

−uij + ujk + uki ≤ 1, ∀ (i , j , k) ∈ V , (13)∑

i,j∈V

uij ≤ L, (14)

uij ∈ {0, 1}, ∀ i , j ∈ V , (15)

vi ∈ {0, 1}, ∀ i ∈ V . (16)

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CCNP - Heuristics

• We modified the MIS heuristic for this problem, but it is easyto create pathological instances.

• We also implemented a Genetic Algorithm for the CC-CNP.

• The GA was able to find optimal solutions for all instancestested.

• Example (again). Here L = 4. Opt Soln = 17.

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Figure: Terrorist network compiled by Krebs.

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Figure: Optimal solution when L = 4.

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Results - Terrorist Network Data

Instance IP Model Genetic Alg ComAlg ComAlg + LSMax Conn. Obj Comp Obj Comp Obj Comp Obj CompIndex (L) Val Time (s) Val Time (s) Val Time (s) Val Time (s)

3 21 188.98 21 0.25 22 0.01 21 0.14 17 886.09 17 0.741 19 0.01 17 0.455 15 30051.09 15 0.871 20 0.18 25 1.3318 − − 13 0.39 14 0.05 13 0.0710 − − 11 0.741 12 0.07 11 0.05

Optimal solution found for all Instances

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Results - Computer Generated Instances

Instance IP Model Genetic Alg ComAlg + LSNodes Arcs Max Conn. Obj Comp Obj Comp Obj Comp

Index (L) Value Time (s) Value Time (s) Value Time (s)20 45 2 9 0.04 9 0.02 9 0.0320 45 4 6 0.13 6 0.04 6 0.86220 45 8 5 0.39 5 0.04 5 1.48225 60 2 11 0.07 11 0.49 11 0.0825 60 4 9 14.1 9 2.113 10 0.0125 60 8 7 26.64 7 0.05 8 0.0630 50 2 11 0.07 11 0.06 11 0.0130 50 4 8 0.1 8 0.05 8 030 50 8 6 1152.15 6 0.09 6 030 75 4 10 18.77 10 0.14 10 0.0230 75 6 9 442.41 9 0.09 9 0.0430 75 10 7 64.94 7 0.18 8 0

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Instance IP Model Genetic Alg ComAlg + LSNodes Arcs Max Conn. Obj Comp Obj Comp Obj Comp

Index (L) Value Time (s) Value Time (s) Value Time (s)35 60 2 12 0.13 12 0.14 12 0.1435 60 4 8 29.89 8 0.711 8 035 60 6 7 31.61 7 0.31 7 0.0140 70 2 15 0.17 15 0.1 15 0.10140 70 4 11 341.97 11 0.06 11 040 70 6 8 78.94 8 0.2 8 0.0445 80 2 16 0.24 16 0.06 16 0.145 80 4 11 48.17 11 0.05 11 0.0245 80 6 8 118.23 8 0.09 8 0.07150 135 2 19 0.36 19 0.27 19 0.0550 135 4 15 165.18 15 0.63 15 0.29150 135 6 14 5722.88 14 0.721 14 0.03

Solution Quality

• GA: optimal solutions found for 100% of cases

• MIS Heuristic: optimal solutions found for 87.5% of cases.

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Instance Genetic Algorithm ComAlg ComAlg + LSNodes Arcs Index (L) Obj Time (s) Obj Time (s) Obj Time (s)

75 140 5 18 1.622 21 0 18 1.50275 140 8 14 1.442 20 0.02 14 1.18175 140 10 12 1.231 20 0.12 12 3.36475 210 5 23 1.532 29 0.01 23 18.47675 210 8 21 2.443 23 0.01 22 2.93475 210 10 20 2.794 24 0.09 20 21.1775 280 5 31 3.464 35 0.101 31 3.14475 280 8 29 2.874 31 0.05 29 3.74675 280 10 28 3.775 30 0.13 28 4.787100 194 5 22 5.317 33 0.02 22 2.774100 194 10 17 3.224 22 0.241 17 6.499100 194 15 15 2.954 22 0.021 15 0.44

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Instance Genetic Algorithm ComAlg ComAlg + LSNodes Arcs Index (L) Obj Time (s) Obj Time (s) Obj Time (s)100 285 5 33 5.08 38 0.02 33 1.262100 285 10 28 4.376 31 0.05 28 11.076100 285 15 27 5.728 28 0.16 27 1.142100 380 5 40 9.052 47 0.051 42 5.739100 380 10 36 11.506 41 0.02 37 3.866100 380 15 35 6.198 40 0.39 36 3.034125 240 5 29 7.951 37 0.251 31 1.472125 240 10 24 9.984 29 0.07 24 1.993125 240 15 22 5.888 26 0.18 22 9.233150 290 5 31 7.981 40 0.421 30 5.798150 290 10 26 4.967 32 0.2 25 5.107150 290 15 23 5.457 29 1.101 23 19.889150 435 5 49 9.143 57 0.06 49 6.459150 435 10 40 19.407 50 0.44 41 5.518150 435 15 38 9.703 45 0.07 38 13.699

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Summary

• Contributions

• K-CNP• Propose math program based on integer linear programming.• Proof of computational complexity• Implement an efficient heuristic based on maximal independent

sets• Heuristic finds optimal solutions for all instances tested in

fraction of time required by CPLEX

• CC-CNP• Math Programming formulation• Genetic Algorithm implemented finds optimal solutions for all

instances tested.

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Summary

Current Work

• Weighted version of the problem

• Approximation of the problem

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References

A. Arulselvan, C.W. Commander, L. Elefteriadou, P.M. Pardalos.Detecting critical nodes in social networks.Computers and Operations Research, 2008.

A. Arulselvan, C.W. Commander, P.M. Pardalos, O. Shylo.Managing network risk via critical node identification.Risk Management in Telecommunication Networks, N. Gulpinar and B. Rustem(editors), Springer, to appear 2008 (in process)

A. Bavelas.A mathematical model for group structure.Human Organizations, 7:16–30, 1948.

S.P. Borgatti.Identifying sets of key players in a network.Computational, Mathematical and Organizational Theory, 12(1):21–34, 2006.

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Outline

1 Introduction





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Single Source and Single SinkMultiple Sources and Multiple SinksSummary

Outline

1 Introduction





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Introduction

Evacuation from Hurricane Katrina

“The live footage of carsstreaming out of the city on

both side of the interstate,with the Superdome and thecity skyline in thebackground, superimposedon CNN’s split-screen withthe mayor and the governorordering everyone out of thecity, is really very, very eerie,strikingly apocalyptic.”

Figure: New Orleans

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Introduction (cont’d)

Evacuation from Hurricane Rita

Figure: I-10

“The Contra FlowEvacuation Of NewOrleans on InterstateTen. The massiveevacuation is full blownwith traffic backed up formiles and moving at acrawl at best. ”

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Maximum Contraflow (MCF)

DefinitionInstance: Given a directed graph G = (V ,A) with source s+ ∈ V ,sink s− ∈ V and capacity ce ∈ Z

+ on each arc e ∈ A.

Question: What is the maximum flow from node s+ to node s−,if the direction of the arcs can be reversed?

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Maximum Contraflow (MCF): Example

s+

n1

n2

n3

n4

s−

1

10

1

31

1

1

1

1

1

3

1

1

2

Figure: Maximum contraflow is 6.

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Maximum Contraflow (MCF): Procedure

1. Construct the transformed graph G = (V , A) where the arc set is defined as

(i, j) ∈ A, if (i, j) ∈ A or (j, i) ∈ A ,

The arc capacity function c is given by

ci,j := ci,j + cj,i ,

for all arcs (i, j) ∈ A.

2. Solve the maximum flow problem on graph G with capacity c.

3. Perform flow decomposition into path and cycle flows of the maximum flow resulting from step2. Remove the cycle flows.

4. Arc (j, i) ∈ A is reversed, if and only if the flow along arc (i, j) is greater than ci,j , or if thereis a non-negative flow along arc (i, j) /∈ A and the resulting flow is the maximum flow with arcreversal for graph G = (V , A).

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Maximum Contraflow (MCF): Complexity

TheoremThe maximum contraflow problem can be solved in stronglypolynomial time.

Corollary

The static version of the maximum contraflow problem withmultiple sources and multiple sinks is strongly polynomiallysolvable.

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Maximum Dynamic Contraflow (MDCF)

DefinitionINSTANCE: Given a directed graph G = (V ,A) with source nodes+ ∈ V , sink node s− ∈ V , capacity ce ∈ Z

+ and transmissiontime te ∈ Z

+ on each arc e ∈ A with ti ,j = tj ,i if (i , j), (j , i) ∈ A,and an overall time horizon T ∈ Z

+.

QUESTION: Determine the maximum amount of flow that canbe send in T units of time from source s+ to sink s−, if thedirection of the arcs can be reversed at time 0.

Note: In this case, if we choose to switch an arc, it remainsswitched from time 0 to T .

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Time Expanded Graph and Temporally Repeated Flows

1

2

3

4

3 , 3

2 , 1 3 , 3

2 , 1

1 , 1 1 , 1

Figure: Graph with Travel time on arcs

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Time Expanded Graph and Temporally Repeated Flows

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

T i m e0 1 2 3 4 5

1

2

3

4

Figure: Time Expanded Graph

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Maximum Dynamic Contraflow (MDCF): Procedure

1. Construct the transformed graph G = (V , A) where the arc set is defined as

(i, j) ∈ A, if (i, j) ∈ A or (j, i) ∈ A .

The arc capacity function c is given by

ci,j := ci,j + cj,i

and the traveling time is

ti,j(= tj,i

):=

{ti,j , if (i, j) ∈ Atj,i , otherwise

,

for all arcs (i, j) ∈ A.

2. Generate a dynamic, temporally repeated flow on graph G with capacity c and traveling time t.

3. Perform flow decomposition into path and cycle flows of the flow resulting from step 2. Removethe cycle flows.

4. Arc (j, i) ∈ A is reversed, if and only if the flow along arc (i, j) is greater than ci,j , or if thereis a non-negative flow along arc (i, j) /∈ A and the resulting flow is the maximum flow with arcreversal for graph G = (V , A).

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Maximum Dynamic Contraflow (MDCF): Complexity

We need a few concepts:

• time expanded graph

• temporally repeated flows

• Theorem of Ford & Fulkerson

TheoremThe maximum dynamic contraflow problem can be solved instrongly polynomial time.

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Dynamic Transshipment Contraflow (DTCF)

DefinitionINSTANCE: A directed graph G = (V ,A), a set of sourcesS+ ⊂ V , a set of sinks S− ⊂ V , arc capacities ce ∈ Z

+ andtransmission time te ∈ Z

+ for each arc e ∈ A with ti ,j = tj ,i if(i , j), (j , i) ∈ A, and an overall positive integer time bound T .

QUESTION: Is there a feasible dynamic flow within time horizonT , allowing each arc to be revered once at time 0?

TheoremDTCF is NP− complete in the strong sense.

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DTCF: Proof of NP− completeness

c+2

+1

c+1

+1

u13 u2

3

u13

u23

u12

u22

u12

u22

u11 u2

1

u11

u21

d−3

−1

d+3

+1

d−2

−1

d+2

+1

d−1

−1

d+1

+1

s−

−2

(2, 3)

(2, 1)

Figure: Transformed graph G3SAT corresponding to 3SAT instance

C = {{u1, u2, u3}, {u1, u2, u3}}

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What makes this problem so tough?

s+2

s+1

n2

n1

s−

(7,3)

(10,1)

(1,1) (1,1)

(2,1)

(2,1)

7

10

-17

s+2

s+1

n2

n1

s−

7

10

1 116

91

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Fixed Switching Cost Contraflow (FSCF): Definition

DefinitionINSTANCE: A directed graph G = (V ,A) with a set of sourcesS+, a set of sinks S−, excess b ∈ Z

|V |, arc capacities ce andarc-switching cost bf

e for each arc e ∈ A.QUESTION: Find a feasible flow f in G with minimal total cost,if the direction of the arcs can be reversed with (fixed) cost bf .

Note that FSCF is a static problem with multiple sources andmultiple sinks. The fixed cost bf

e occur, whenever arc e is reversed.

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Fixed Switching Cost Contraflow (FSCF): Complexity

Equivalent problem:

• 0/1-Minimum Improvement Flow (MIF)

• Complexity discussed by Krumke et al. in 1998

TheoremFSCF is NP− complete in the strong sense.

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Summary

MCF problem is polynomial solvable

MDCF problem is polynomial solvable

DTCF is NP− complete

FSCF is NP− complete

Future Work: Solution approaches to the problem

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References

S. Rebennack, A. Arulselvan, L. Elefteriadou, P. M. Pardalos.Complexity Analysis for Maximum Flow Problems with Arc Reversals.Submitted to Networks, 2008.

FL. R. Ford and D. R. Fulkerson.Flows in Networks.Princeton University Press, Princeton, New Jersey, 1962.

B. E. Hoppe.Efficient Dynamic Network Flow Algorithms.PhD thesis, Cornell University, 1995.http://www.math.tu-berlin.de/~skutella/hoppe_thesis.ps.gz.

R. K. Ahuja, T. L. Magnati, and J. B Orlin.Network Flows: Theory, Algorithms, and Applications.Prentice Hall, Englewood Cliffs, New Jersey, 1993.

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http://www.math.tu-berlin.de/~skutella/hoppe_thesis.ps.gz





References (cont’d)

S. Krumke, H. Noltemeier, S. Schwarz, H. Wirth, and R. Ravi.Flow Improvement and Network Flows with Fixed Costs.In In Proceedings of the International Conference of Operations Research Ziirich(0R’98), pages 158–167, 1998.

G. M. Guisewite and P. M. Pardalos.Minimum Concave-Cost Network Flow Problems: Applications, Complexity, andAlgorithms.Annals of Operations Research, 25:75–100, 1990.

D. Kim and P. M. Pardalos.Dynamic Slope Scaling and Trust Interval Techniques for Solving ConcavePiecewise Linear Network Flow Problems.Networks, 35(3):216–222, 2000.

S. Kim and S. Shekhar.Contraflow Network Reconfiguration for Evaluation Planning: A Summary ofResults.In Proceedings of the 13th annual ACM international workshop on Geographicinformation systems, pages 250–259, 2005.

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IntroductionFormulationBranch and Price

Outline

1 Introduction





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Introduction

• A bimodal evacuation Problem

• Modes of transportation• Buses• Cars

• Line Planning Problem

• Time expanded graphs

• Goals• Frequency of bus routes• Paths of cars

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Problem Definition

DefinitionGiven a graph G (V ,E ) with cij , uij and tij as the cost, capacityand travel time respectively on each arc (ij) ∈ E , a set of T bustours, and two sets of origin destination pairs (OD)2 and (OD)1corresponding to demands of origin and destinations of cars andpeople traveling by bus respectively, determine the minimum costpath of the cars and frequency of the bus routes satisfyingdemands and capacity.

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Assumptions

• Static demands and travel time

• Routes of buses already known

• Zero loading and unloading time

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Formulation and Discussion

Maximize∑

p∈P1

cpxp +∑

p∈P1

cp fp (17)

s.t. ∑

p1∈P1

γ1(st, p)xp = d1st , ∀st ∈ OD1 (18)

∑

p2∈P2

γ2(st, p)fp = d2st , ∀s ∈ OD2 (19)

∑

p∈P1

α1(ij, p)xp −∑

t∈T

β(ij, t)Bbt ≤ 0, ∀(ij) ∈ E (20)

∑

p∈P2

α2(ij, p)fp +∑

t∈T

β(ij, t)bt ≤ cij , ∀(ij) ∈ E (21)

xp ∈ Z+, ∀p ∈ P1 (22)

fp ∈ Z+, ∀p ∈ P2 (23)

bt ∈ {0, 1}, ∀t ∈ T (24)

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Branch and Price

• Restricted Master Problem (RMP)

• Subproblems

• Branching Strategies

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Restricted Master Problem

Replacing the sets P1 and P2 by the restricted sets P′1 ⊆ P1 and

P′2 ⊆ P2 respectively and relax the integer constraints.

Maximize∑

p∈P′1

cpxp +∑

p∈P′1

cp fp (25)

s.t. ∑

p1∈P′1

γ1(st, p)xp = d1st , ∀st ∈ OD1 (26)

∑

p2∈P′2

γ2(st, p)fp = d2st , ∀s ∈ OD2 (27)

∑

p∈P′1

α1(ij, p)xp −∑

t∈T

β(ij, t)Bbt ≤ 0, ∀(ij) ∈ E (28)

∑

p∈P′2

α2(ij, p)fp +∑

t∈T

β(ij, t)bt ≤ cij , ∀(ij) ∈ E (29)

xp ≥ 0, ∀p ∈ P′

1 (30)

fp ≥ 0, ∀p ∈ P′

2 (31)

0 ≤ bt ≤ 1, ∀t ∈ T (32)

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RMP (cont’d)

• Let π ∈ ROD1 be the unrestricted dual variable corresponding

to constraint set (26)

• µ ∈ ROD2 be the unrestricted dual variable corresponding to

constraint set (27)

• η ∈ RE− be the non-positive dual variable corresponding to the

constraint set (28)

• ψ ∈ RE− be the non-positive dual variable corresponding to the

constraint set (29)

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People-Path Subproblem

In the people-path subproblem, we determine the minimum reducedcost, xp, of a path flow variable, xp, for people taking busesbetween a given origin-destination pair st ∈ OD1. This is given by

xp = cp − (πst +∑

∀(ij)∈E

α1(ij , p)ηij ) = −πst +∑

∀(ij)∈E

(α1(ij , p)cij − α1(ij , p)ηij ) (33)

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Car-Path Subproblem

In the car-path subproblem, we are concerned with the reducedcost, fp, of car flow variables, fp, for the pairs of origin-destinationst ∈ OD2. This given by

fp = cp − (µst +∑

∀(ij)∈E

α2(ij , p)ψij ) = −µst +∑

∀(ij)∈E

(α2(ij , p)cij − α2(ij , p)ψij ) (34)

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Branching Strategies

• Always branch on variable bt , if fractional

• No paths to enter and all bt are integer and there arefractional fp or xp variables

• Choose two fractional paths p1 and p2 between the same originand destination pair

• Find the node i at which they diverge• Keep the set of arcs A1(ij) ⊂ A(ij) forbidden in one branch

and A2(ij) = A(ij)\A1(ij) forbidden in the other branch

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Reference

C.Barnhart, C.A. Hane, and P.H. Vance.Using branch-and-price-and-cut to solve origin-destination integermulticommodity flow problems.Oper. Res., 48(2):318–326, 2000.

J.H.M. Goossens, C.P.M. Hoesel, and L.G. Kroon.On solving multi-type line planning problems.METEOR Research Memorandum RM/02/009, University of Maastricht,Maastricht, The Netherlands, 2002.

M.E. Pfetsch and R.Borndrfer.Routing in line planning for public transport.In Herbert Kopfer Hans-Dietrich Haasis and Jrn Schnberger, editors, OperationsResearch Proceedings, pages 405–410. 2005.

F. Alvelos.Branch-and-Price and Multicommodity Flows.PhD thesis, Universidade do Minho, 2005.http://repositorium.sdum.uminho.pt/dspace/bitstream/1822/2736/1/falvelos_phd.pdf.

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http://repositorium.sdum.uminho.pt/dspace/bitstream/1822/2736/1/falvelos_phd.pdf

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