NetworkPartitioningAlgorithmsforSolving …sites.utexas.edu/cm2/files/2018/05/NetworkPartitioning... · 2018. 4. 26. · I SDDA generates partitions with a lower number of boundary

Department of Civil, Architectural, & Environmental Engineering

The University of Texas at Austin

Network Partitioning Algorithms for Solvingthe Traffic Assignment Problem using a

Decomposition Approach

Cesar N. Yahia, Venktesh Pandey, & Stephen D. BoylesCM2

April 26, 2018

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Content

Background & MotivationTraffic & CongestionTraffic AssignmentA Decomposition Approach for the Static Traffic Assignment Problem

Network PartitioningObjectivesPartitioning AlgorithmsPartitioning Results

Conclusions

Cesar N. Yahia, Venktesh Pandey, & Stephen D. Boyles | Network Partitioning Algorithms for Solving the Traffic Assignment Problem using a Decomposition Approach

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Background & MotivationTraffic & Congestion

I As megaregions grow and become more interconnected, the flow ofgoods and people across megaregions increases

I This creates traffic management challenges for the differentmetropolitan planning organizations involved within the megaregion

Figure: commuters and traffic across megaregions (Routley, 2017)


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Background & MotivationTraffic Assignment

I The traffic assignment problem (TAP) is used to predict driver routechoice and link flows (congestion) for a given level of travel demand

I The static version of the problem is commonly employed by planningagencies, and it can be formulated as a convex program

I Modern specialized algorithms can solve this problem efficiently

Figure: The traffic assignment problem as a convex optimization problem(Beckmann, 56)


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Background & MotivationTraffic Assignment

I Benefits of solving the traffic assignment problem for an entiremegaregion include:

1. determining congestion levels on transport links connecting megaregioncities

2. identifying dependencies between congestion states in differentmegaregion cities

3. determining the impact of policies on travelers from different originsacross the megaregion

I However, such large instances of the traffic assignment problem arecomputationally challenging to solve using traditional specializedalgorithms

I Therefore, there is a need to parallelize the traffic assignment problemacross subnetworks within the megaregion


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Background & MotivationA Decomposition Approach for the Static Traffic Assignment Problem (DSTAP)

I DSTAP is a spatial parallelization scheme for the static trafficassignment problem

I The network is divided into subnetworks that correspond to trafficassignment subproblems solved in parallel

I The algorithm alternates between equilibrating the subproblems and amaster problem that represents a simplified version of the full network


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1. The full network is divided intoa simplified master problem(left) and several detailedsubproblems (right)

2. The algorithm alternatesbetween equilibrating flows inthe master network and inparallel across subproblems

3. The convergence of thealgorithm to the globalequilibrium across the fullnetwork is proved in (Boyles,2017)

4. This global equilibriumaccounts for effects betweensubnetworks and determinesflows on links connectingsubnetworks


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I the algorithm relies onresults from equilibriumsensitivity analysis toapproximate changes intraffic within a subnetworkusing aggregate links(Boyles, 2012)


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Computation time at each iteration dependson:1. The aggregate links generated using

equilibrium sensitivity analysis torepresent subnetworks

2. The time needed to solve TAP on thesubproblems in parallel which isdominated by the largest subproblem

Computation time needed to convergetowards global equilibrium depends on theinter-flow between subnetworks�OD ≤ 2B(�M + �S)


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Network PartitioningObjectives

I The computational performance of DSTAP critically depends on theboundary of the subnetworks since this determines:

1. The number of aggregate links for each subnetwork2. The size of each subnetwork equilibrium subproblem3. The inter-flow between subnetworks

I Additionally, the region associated with each metropolitan planningagency should correspond to a distinct subnetwork

1. This is desired since different planning agencies and DOT’s do not have acommon platform for sharing network data relevant to traffic assignment

2. The planning agencies may also be very different in terms of technologyused to run traffic assignment for their own transportation network


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Network PartitioningObjectives

Therefore, the objective of this research is to develop partitioningalgorithms that result in subnetworks which1. optimize the computational performance of DSTAP2. enhance the application of this algorithm considering the needs of

distinct metropolitan planning agencies in a megaregion


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Network PartitioningPartitioning Algorithms

Algorithm 1:Shortest domain decomposition algorithm (SDDA) (Johnson et al., 2016)

I An agglomerative heuristic that aims to minimize boundary nodes as aprimary objective and to maintain balanced partitions


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Network PartitioningPartitioning Algorithms

Algorithm 2:Flow-weighted spectral partitioning

I This algorithm partitions the network based on the second smallesteigenvalue of the flow-weighted normalized graph Laplacian

I This minimizes the inter-flow between subnetworks and createssubnetworks that are balanced by total flow

Lsymm = D−1/2G LGD

−1/2G (1)

LGϕi = λiϕi (2)

λi = minS of dim i

maxx∈S

xTLGxxTx

(3)

ϕi = arg minS of dim i

maxx∈S

xTLGxxTx

(4)


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Network PartitioningPartitioning Results

Computation time per DSTAP iteration

Network Boundary nodes Time (s)Austin (SDDA) 174 632.83

Austin (Spectral) 329 746.81Austin (4 subnets, SDDA) 296 290.97Austin (4 subnets, spectral) 440 82.50

Anaheim (SDDA) 46 0.10Anaheim (Spectral) 48 0.13

Chicago sketch (SDDA) 74 9.79Chicago sketch (Spectral) 50 7.42


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Computation time per DSTAP iterationI SDDA generates partitions with a lower number of boundary nodesI Flow balanced partitions are important for parallelizing subproblems.

For Austin (4 partitions), SDDA subproblems require 3.5 times morecomputation time due to the flow imbalance


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DSTAP convergence rate

Network Inter-flowAustin (SDDA) 186161

Austin (Spectral) 137940Austin (4 subnets, SDDA) 368718Austin (4 subnets, spectral) 296870

Anaheim (SDDA) 81991Anaheim (Spectral) 56539

Chicago sketch (SDDA) 154791Chicago sketch (Spectral) 201603


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DSTAP convergence rate

I Spectrial partitioning generatessubnetworks with lower inter-flow

I For a hypothetical network of two SiouxFalls networks connected by lowdemand, DSTAP converges faster withspectral partitioning subnetworks


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Separating MPO’s into distinct subnetworks

I The full Texas network wasdivided into two subnetworksusing SDDA (left) &flow-weighted spectralpartitioning (right)

I The cut generated by SDDAcrosses through the Austinarea, dividing Austin into twoportions across eachsubnetwork

I The cut generated by thespectral partitioning algorithmkeeps the region controlled bya planning agency intact withinone subnetwork


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Conclusions

I Traffic assignment across a megaregion has significant benefits forplanning, but solving this problem can be computationally challenging

I An efficient parallelization scheme that determines a globalequilibrium solution was developed (DSTAP)

I Computation time per DSTAP iteration can be reduced by minimizingboundary nodes and balancing subproblems

I SDDA generates partitions with a lower number of boundary nodesI Flow-weighted spectral partitioning generates subproblems with better

balance


19Conclusions

I DSTAP convergence rate depends on inter-flow between subnetworksI Flow-weighted spectral partitioning generates subnetworks with lower

inter-flowI For a hypothetical network of two Sioux Falls networks, DSTAP converges

faster with flow-weighted spectral partitioning subnetworksI Assigning a planning agency to one distinct subnetwork is critical for a

parallelization algorithmI Flow-weighted spectral partitioning for Texas did not divide any planning

agency across subnetworksI SDDA partitions for Texas cut through the Austin area


Thank you! Questions?

Background & MotivationTraffic & CongestionTraffic AssignmentA Decomposition Approach for the Static Traffic Assignment Problem

Network PartitioningObjectivesPartitioning AlgorithmsPartitioning Results

Conclusions

Documents

NetworkPartitioningAlgorithmsforSolving …sites.utexas.edu/cm2/files/2018/05/NetworkPartitioning... · 2018. 4. 26. · I SDDA generates partitions with a lower number of boundary