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A network partitioning methodologyfor distributed traffic managementapplicationsHamideh Etemadniaa, Khaled Abdelghanya & Ahmed Hassana

a Bobby B. Lyle School of Engineering, Southern MethodistUniversity, PO Box 750340, Dallas, TX 75275-0340, USAAccepted author version posted online: 22 Apr 2013.Publishedonline: 15 Jul 2013.

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Transportmetrica A: Transport Science, 2014Vol. 10, No. 6, 518–532, http://dx.doi.org/10.1080/23249935.2013.795200

A network partitioning methodology for distributed trafficmanagement applications

Hamideh Etemadnia1*, Khaled Abdelghany and Ahmed Hassan

Bobby B. Lyle School of Engineering, Southern Methodist University, PO Box 750340,Dallas, TX 75275-0340, USA

(Received 31 January 2012; final version received 9 April 2013)

This paper presents a multi-way network partitioning methodology for distributed traffic man-agement applications. The methodology can be used to partition a typical urban transportationnetwork such that: (a) the inter-flow among the resulting subnetworks is minimised; (b) thesubnetworks are balanced in terms of their sizes/flow activities; and (c) each subnetwork isconnected. Two heuristics are presented. The first adopts a recursive iterative procedure todetermine the network’s sparsest cuts that maintain the balance and connectivity requirements.The second heuristic adopts a greedy network coarsening technique to determine the mostflow-independent subnetworks. The solution quality of these two heuristics is evaluated usinghypothetical and real networks with different configurations. The results show that the heuristicscan obtain near-optimal solution in significantly shorter execution times.

Keywords: distributed traffic management; network partitioning; network coarsening;sparsest cut

Introduction

Urban areas in the USA and in many countries around the world have rapidly grown in sizeand population resulting in the formation of mega-cities with congested transportation networks.For instance, the Los Angeles (LA) metropolitan area in California has grown at a rate of about35% from 1982 to 2007 recording a total population of 17.8 millions living in 189 cities thatextend over an area of 4850 square miles. With an average annual delay of 70 h per traveller, theLA metropolitan area is ranked as the most congested urban area in the USA (Schrank and Lomax2009). Traffic management in such large congested urban areas is considerably challenging. Itrequires the development of traffic management capabilities that effectively alleviate recurrentand non-recurrent congestion. A holistic approach to develop these capabilities would follow acentralised architecture through which the entire traffic network is managed using one centralcontroller. This controller is equipped with the capabilities to collect real-time information onthe current congestion pattern across the entire network, generate a suitable traffic managementscheme, and disseminate this scheme to the traffic control devices for implementation (e.g. traf-fic light, dynamic message signs, traveller information, etc.). However, the large size of theseurban areas represents a barrier in face of adopting the centralised architecture. It requires an

*Corresponding author. Email: hetemadnia@mail.smu.edu1Present affiliation: Department of Agricultural Economics, Sociology and Education, Pennsylvania StateUniversity, University Park, PA 16802-5602.

© 2013 Hong Kong Society for Transportation Studies Limited

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extensive wired and wireless communication infrastructure to interconnect these large areas. Fur-thermore, the large network sizes make it algorithmically difficult to develop a global optimaltraffic management scheme, especially if this scheme needs to be generated in real-time (Peetaand Ziliaskopoulos 2001).

A plausible alternative approach is to adopt a decentralised traffic management architecture. Thearchitecture builds on an existing infrastructure for data collection and traffic management withineach local jurisdiction. Extensive research work has been devoted to describe the framework andmethodological aspects of the decentralised traffic management architecture (Cuena, Hernandez,and Molina 1995; Hawas and Mahmassani 1995; Pavlis and Papageorgiou 1999; Hernandez,Ossowski, and Garcia-Serrano 2002; Logi and Ritchie 2002; Daganzo 2007). For example, Hawasand Mahmassani (1995) proposed a decentralised architecture in which the network is managedby multiple distributed controllers. Each controller manages traffic within a pre-defined subarea.The route guidance instructions and other control strategies are generated by each controller usingtraffic surveillance data for the portion of the network within the controller’s locality. Each localcontroller is assumed to estimate the network conditions outside its boundaries based on availablehistorical information. Chiu and Mahmassani (2002) present a hybrid framework that integratesthe centralised and the decentralised approaches. Travellers are split in terms of their source ofroute guidance instructions (centralised versus decentralised). The centralised controller providesroute guidance instructions based on the predicted network conditions, while the decentralisedcontrollers provide route guidance instructions that react to the actual traffic conditions withinthe locality of each controller. Also, Hernandez, Ossowski, and Garcia-Serrano (2002) present aknowledge-based approach to develop a distributed architecture in which a structural cooperationmechanism is used to model the coordination among local controllers.

Nonetheless, one should expect the quality of a traffic management scheme using a distributedarchitecture to depend primarily on the boundaries of the local controllers, and their interfaces withthe main traffic movements in the network (i.e. traffic hand-off points). Most existing researchwork that adopts distributed traffic management architectures seems to overlook such dimen-sion of the problem assuming that the boundaries of the local controllers are predefined. In arecent research work (Wen 2009; Villalobos, Chiu, and Mirchandani), network decompositionis proposed as a strategy to achieve scalable dynamic traffic assignment (DTA) implementation.To reduce the computational complexity of the existing DTA methodologies, Wen (2009) andVillalobos, Chiu, and Mirchandani propose solution techniques to improve the scalability of DTAmodels for on-line applications. Boundary construction and adjustment algorithm are proposed inJi and Geroliminis (2011, 2012) to obtain partitioning results such that each subnetwork has well-defined macroscopic fundamental diagrams. Several criteria should be considered to determinethe boundaries of the distributed controllers in the decentralised architecture. For example, thenetwork should be partitioned such that the subnetworks are flow-independent. In other words,the partition minimises inter-flow among adjacent subnetworks in order to reduce the need fortraffic management schemes that require intensive communication/coordination among adjacentcontrollers. In addition, the network partitioning should maintain a balance in terms of the amountof traffic management activities required by each local controller. For instance, the partitioningshould be conducted such that the subnetworks are relatively close in size (spatial balance), and/orthey are similar in terms of the amount of served traffic within their boundaries (flow balance). Thebalance criterion would likely enable the decentralised architecture to achieve the real-time com-putational requirement as it distributes the computation effort, associated with developing thetraffic management activities, equally among controllers. Furthermore, the partitioning shouldensure the spatial connectivity of each subnetwork. The boundaries of each subnetwork shouldenvelope all control devices (e.g. traffic lights) that are operated by the controller to facilitatecommunication among these control devices, especially if wireless communication is used.

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In this paper, we investigate the problem of network partitioning for distributed traffic man-agement applications. We present a methodology in which the network is divided such that theabove-mentioned criteria are considered. As the problem is proved to be nondeterministic poly-nomial time-hard (NP-hard), two heuristics are developed to solve it efficiently. The first heuristicrecursively determines a near-optimal sparsest cut through solving the maximum concurrent flow(MCF) problem while maintaining the balance and connectivity requirements. The second heuris-tic adopts a greedy-based network coarsening technique to determine the most flow-independentsubnetworks. The paper contributes to the existing literature by providing a mathematical formu-lation to the traffic network partitioning problem. In addition, the presented methodology is novelas it simultaneously ensures independency and load balance of the obtained partitions. The paperis organised as follows. A literature review of the network partition problem is presented in thenext section. A formal definition of the network partitioning problem in the form of an integermathematical program is presented. We then describe the two heuristics that are developed tosolve this problem. The results of a set of experiments that illustrate their performance are alsopresented. Finally, conclusions and planned research extensions are discussed.

Literature review

Considering its numerous applications, several versions of the network partitioning problem havebeen studied in the graph theory literature (Shmoys 1997). For example, the minimum multicutproblem determines the minimum cost cut in a network such that the source and the sink nodes ofany predefined source–sink pair are located in two different partitioned components. The sparsestcut problem determines the minimum density cut in the network. It determines the cut with theminimum ratio between the cost of the cut and the total demand between source–sink pairs thatare disconnected due to this cut. The α-balanced cut problem aims at determining the minimumcost cut by separating the network into two components such that one component includes αnnodes, where n is the number of nodes in the network. The bisection cut problem is a specialcase of the α-balanced cut where α is equal to 0.5. The network partitioning problem, includingthe versions mentioned above, has been proven to be NP-hard (Garey and Johnson 1979; Matulaand Shahrokhi 1990). Therefore, an approximate solution that is based on solving a relaxed linearapproximation of the problem is proposed (Matula and Shahrokhi 1990). For example, the dualof the linear approximation of the sparsest cut problem is the MCF problem. Linial, London,and Rabinovich (1995) and independently Garg, Vazirani, and Yannakakis (1996) have shownthat these approximated algorithms could run in polynomial expected time, and are guaranteed toproduce a cut of sparsity ratio of O(log k) factor of the optimal solution, where k is the numberof source–sink pairs in the network. Rao (1987) emphasised the importance of the sparsest cutin computing a balanced cut. For any cut S, the sparsity ratio is equal to the ratio of: (I) the totalcost of the cut (e Ce, where e is the list of links in the cut S, and Ce is the cost of link e);nd(II) the total disconnected demand (k df , where k is a disconnected source–sink pair and df isthe demand of this pair). By solving the special case of the sparsest cut problem in which weconsider ( n

2 ) source–sink pairs, and setting the demand between every pair to be one unit (theall pairs unit-demand sparsest cut problem), the denominator of the sparsity ratio is maximisedby having two separated components of roughly equal size which results in a balanced cut. Inorder to achieve a cut that is as close as possible to the required α-balance value, an iterativeheuristic approach that adopts a greedy-ratio strategy is proposed. This greedy-ratio heuristic hasbeen proved to find an optimal α-balanced cut for any α ≤ 1

3 (Shmoys 1997).Another approach that adopts a multi-level recursive heuristic is proposed by Hauck and Bor-

riello (1995), Karypis, Kumar, and Shekhar (1999), Alpert, Huang, and Kahng (1997), Wichlundand Einar (1998) and Karypis and Kumar (2000). The heuristic recursively creates a coarser

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version of the network through combining its nodes to reduce the network size. The coarsen-ing step is conducted such that a predefined global objective is achieved. For example, for acost-minimisation partitioning, nodes that are connected with high-cost links are combined torepresent a coarser node in the next iteration. The remaining least cost links become candidatesfor the cut. A refining strategy could also be used to modify the boundaries obtained from thecoarsening step (Karypis and Kumar 2000). Examples of network partitioning methodologiesdeveloped to target the data clustering problem can be found in Zahn (1971), Johnson (1982),Asano et al. (1988), Xu, Olman, and Xu 2001, Jain and Dubes (1988), Gersho and Gray (1992),and Inaba, Katoh, and Imai (1994). For example,Asano et al. (1988) present a clustering techniquethat uses the minimum spanning tree of a network that defines the similarity between the differentdata points. Links of this tree are first sorted in an ascending order. Then, the first k − 1 linkswith minimum weights are deleted resulting in k partitions. The k-mean variance-based clusteringis another technique that is used to construct k clusters (Inaba, Katoh, and Imai 1994). First, aset of k nodes in the network, called centre points, are defined. The k-mean clustering algorithmattempts to create k partitions by minimising the mean squared of the distance (weight) from eachnode in the network to the nearest centre. A comprehensive and more recent survey of the k-meanclustering algorithm can be found in Jain (2010). Widely known as Lloyd’s algorithm (Lloyd1982), the centre node of each cluster is replaced by the centroid of the cluster and the clusternodes are updated accordingly. The process is repeated iteratively until convergence is achieved,where the location of the centroid and the cluster members stabilises in two successive iterations.Based on this brief survey, existing methodologies could not be directly applied for the problemconsidered in this paper. For example, the application of α-balanced sparsest cut technique islimited only for the network bisection problem. As to the authors’ knowledge, the extension ofthis technique for multi-way balanced network partitioning is not considered. In addition, mostcoarsening algorithms adopt network bisection techniques to split the coarsened network. Thus,these algorithms are suitable only when an even number of partitions are required.

The multi-way balanced network partitioning problem

Given is a highway network G(N , A), where N is the set of nodes and is the set of links. Thenodes in the network can be either intersections or intermediate nodes in long roads to shape thegeometry of the road in the network. The network is assumed to be managed by a set of distributedcontrollers C. A controller is a subnetwork including a subset of nodes and links. Each controllerc ∈ C manages the traffic in its own subnetwork Gc(Nc, Ac), where Nc is the set of nodes and Ac isthe set of links in this subnetwork. Define fij as the vehicular flow during the horizon of interest forlink (i, j). We define the binary decision variable xic which is equal to one if intersection belongsto subnetwork Gc(i ∈ Nc), and zero otherwise. Also, we define the binary decision variable yijc

which is equal to one if link (i, j) belongs to subnetwork Gc and zero otherwise. The problem isto determine the optimal partitioning of the network into a predefined number of subnetworkssuch that: (1) each subnetwork is connected; (2) the inter-flow across the boundaries of thesesubnetworks is minimised; and (3) the flow/spatial balance constraints among the controllersare satisfied. The subnetworks are considered to be disjoint components with minimum interflow dependency. The mathematical program given below is used to formulate this problem. Theobjective function minimises the flow on the cut links as illustrated in Equation (1). The integerdecision variable zij defines the assignment of the links to the different subnetworks. It is equal toone for non-cut links where both intersections i and j belong to the same subnetwork (i ∈ Nc andj ∈ Nc), and is equal to two if a link is a cut link (i ∈ Nc and j ∈ Nc′ where c = c′). Constraintsin Equation (2) ensure that each intersection belongs to only one subnetwork. Constraints inEquations (3) and (4) relate the decision variables X, Y , and Z . In Equation (5), the constraints

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ensure the connectivity of each subnetwork. We define the parameter bmij which is equal to oneif intersection falls in the imaginary rectangle where intersections i and j are two corners alongits diagonal, and m is connected to any of the intersections in the subnetwork that contains iand j. To ensure that the network is not disjoint, if intersections i and j belong to one controllerbmij = 1, intersection m is set to be part of the subnetwork Gc. Constraints in Equations (6) and(7) maintain the balance requirements. Constraints in Equation (6) are used in the case a spatialbalance is required, while constraints in Equation (7) are used for the flow balance case. Theparameter δ and γ define the maximum acceptable difference in the sizes and served flows ofthe resulting subnetworks. The maximum acceptable difference in sizes and flows depends on thenetwork spatial distribution. They can be greater than or equal to 0 and less than and equal to theminimum required number of nodes and the minimum amount of flows in each controller (|Nc|and i

i =jjfij · yijc, respectively). Constraints in Equations (8) and (9) state the binary conditions

of the decision variables xic and yijc, respectively. The decision variable zij is 1, 2 integer asshown in constraints (10).

Minimise ii =j

jfij ∗ (zij − 1), (1)

Subject to: cxic = 1 ∀i ∈ N , c ∈ C, (2)

2 ∗ yijc ≥ xic + xjc ∀(i, j) ∈ Nc, i = j, c ∈ C, (3)

zij = cyijc ∀(i, j) ∈ Nc, i = j, c ∈ C, (4)

xmc ≥ bmij ∗ (xic + xjc − 1) ∀(m, i, j) ∈ Nc, i = j, c ∈ C, (5)

|N ||c| − δ ≤ ixic ≤ |N |

|c| + δ ∀c ∈ C, (6)

ii =j

jfij

|C| − γ ≤ ii =j

jfij · yijc ≤ i

i =jjfij

|C| + y ∀c ∈ C, (7)

yijc ∈ (0, 1) ∀(i, j) ∈ N , i = j, ∀c ∈ C, (8)

xic ∈ (0, 1) ∀i ∈ N , c ∈ C, (9)

zij ∈ 1, 2 ∀(i, j) ∈ N , i = j. (10)

Solution approach

As mentioned above, the network partitioning problem is proved to be NP-hard. In this section, wepresent two heuristics to efficiently solve this problem. We consider the case in which the networkis required to be partitioned into -spatially balanced subnetworks with a balance ratio α = 1

k .Figure 1 illustrates the main steps of the first heuristic (sparsest cut heuristic), which consists oftwo main iterative loops. The outer loop recursively solves the balanced cut problem for differentvalues of the balance ratio α, while the inner loop applies a greedy procedure to obtain the balancedcut for a specific value of α. The cut resulting from the first iteration of the outer loop divides thenetwork into two components with 1

k balance ratio, where the smaller component is one of the krequired partitions. In the next iteration, the remaining part of the network is partitioned using avalue of α that is equal to 1/(k − 1) (i.e. α = 1/(k − 1)). The process is recursively applied untilthe remaining part of the network is divided into two components (α = 0.5), implying that all kpartitions are obtained. The inner loop is based on the work described in Shmoys (1997), whichimplements an iterative greedy procedure to obtain the balanced cut for a given α. In each iteration,the all pairs unit-demand sparsest cut problem is solved to determine the minimum density cut (i.e.

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Figure 1. The α-balanced multi-way network partitioning heuristic.

two components with minimum flow interface). However, as mentioned above, the sparsest cutproblem is closely related to the all-pairs MCF problem. When the solution of the MCF problempartitions the network into two to four components, a sparsest cut can be identified (Matula 1985).

In the current implementation, a linear program is developed to solve the MCF problem. Giventhe solution of this linear program, the list of critical links (i.e. saturated links with positive dualvalue) is identified to define the sparsest cut and associated network components. Comparing thesizes of the resulting components, if the α-balance ratio is satisfied, the smaller component isreported as one partition, and the outer loop is again activated to partition the remaining part ofthe network with an updated value of α. Otherwise, two cases could be encountered as illustratedin Figure 2: (a) the size of the smaller component is less than αn (i.e. more nodes need to beadded to the smaller component to satisfy the α-ration) and (b) the size of the smaller componentis greater than αn (i.e. nodes need to be removed from the smaller component until the α-ratio issatisfied). In case (a), the sparsest cut problem is again solved considering the larger one of the

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< an

an

an an

an

an

<

=

>

>

=

Case “a” Case “b”

Iteration i

Iteration i+1

Final Solution

Figure 2. The greedy procedure for α-balanced network partitioning (the inner loop in Figure1).

two components that are obtained in the previous iteration. The resulting smaller component fromthe current partitioning is added to the smaller component of the previous iteration, if they arespatially connected. If they are not connected, the larger component from the current partitioningis added instead to the smaller component of the previous iteration to ensure connectivity. Theprocess continues until the α-ratio is satisfied, and the union of the added components in alliterations is reported as the subnetwork for this value of α. In case (b), the sparsest cut problem issolved considering the smaller of the two subnetworks that are obtained in the previous iteration.The size of the smaller component from the current iteration is checked against the requiredα value. The process is repeated until the α-balanced condition is satisfied. Then, the smallersubnetwork is reported as one partition and all the remaining components are grouped as onenetwork for partitioning in the next iteration of the outer loop. It should be noted that, in case (b),the optimal sparsest cut that is used to determine the portion of the network to be eliminated, tosatisfy the balance condition, might result in disconnecting these nodes from the remaining part ofthe network. The heuristic treats this situation by arbitrarily modifying the optimal cut to ensurethat the eliminated portion is still connected with the remaining part of the network. In addition,if the flow balance is to be considered instead of the spatial balance, the subnetwork’s intra-flowpercentage will be compared with the ratio. All other steps are unchanged. An efficient algorithmto determine the sparsest cut could have a complexity of O(log n) (Rao 1987). However, to ensurethat the inner loop step of the heuristic described above is producing the required α-balanced cut,this step could be repeated n times, in the worst case, resulting in a complexity of O(n log n). Thus,to obtain the k balanced partitions, a worst-case complexity of O(nk log n) could be recorded.

The second heuristic adopts a network coarsening approach similar to the one presented inKarypis and Kumar (2000). However, the new heuristic aims at maintaining the balanced require-ments for any number of partitions (even or odd). It also simultaneously generates all subnetworksrather than solving a network bisection problem at each iteration. Given the number of requiredpartitions, the coarsening heuristic starts by determining a centroid node for each partition. Dif-ferent strategies could be used to select the centroid nodes. For example, they could be manuallyidentified based on the analyst’s knowledge of the network topology. Alternatively, a search pro-cess could be activated to identify nodes with the highest flows (i.e. the sum of the flows of theiroutbound links) and spatially distributed to cover the entire network. Starting from a centroidnode and adopting a greedy heavy link matching strategy, the adjacent node with the maximumlink flow is combined with the centroid forming a coarser centroid. The network is then updatedto provide the next-level coarser network, as illustrated in Figure 3. To maintain the spatial and

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Centroid

Node A

Before combining node A After combining node A

Figure 3. Example of the heavy link matching strategy and network updating.

flow balance among the partitions, this process is sequentially rotated among all centroids. Theheuristics terminates when every node in the original network is combined with a centroid noderesulting in the final coarsened network. This coarsened network has a number of nodes equalto the number of required partitions. Each node in the final network represents a partition. Thiscoarsened node is projected back to retrieve its subnetwork. Two main steps constitute this heuris-tic which are: (a) node scanning for coarsening and (b) determining the heaviest link at each noderesulting in a worst-case complexity of O(n log a), where a = |A|.

Results and analysis

The two heuristics described above are implemented in Java, and the Java-based ILOG Concerttechnology is used to solve the MCF linear program of the first heuristic (ILOG CPLEX 10.02006). A desktop machine with 3.0 GHz and 3.0 GB RAM is used to run the two programs. Aset of experiments are conducted to compare the performance of the two heuristics to that ofthe optimal solution. The optimal solution is obtained through solving the mathematical programdescribed above, which is also implemented using the Java ILOG technology. We first consider agrid hypothetical network with a randomly generated link flows following a uniform distributionu[0, 1000]. Table 1 illustrates the resulting subnetworks and the corresponding execution times forthe heuristics and the optimal mathematical program. The flow assigned to each link is indicatedby its thickness as illustrated in the given network sketches. The network is partitioned into 2, 4, 5,8 and 10 subnetworks. For each case, we record the total intra-flow and inter-flow as percentagesof the total link flows. The intra-flow percentage is the sum of the flows of all links that are notpart of any of the cut links, which is computed as the percentage of the network’s total link flows.The percentile ratio between the sum of the link flows of all cut links and the total network flow isthe inter-flow percentage. To measure the spatial balance, the coefficient of variation (CoV) withrespect to the number of nodes in the different subnetworks is recorded. Similarly, the CoV of thesubnetworks’ link flows is recorded to measure their flow balance.

As shown in Table 1, the two heuristics achieve a good partitioning performance at less runningtime compared with the optimal solution. It is worth mentioning that due to extensive runningtime, the results for the optimal solution for the 5, 8, and 10 partition cases are recorded after a2 h (7200 s) running time before it produces the optimal solutions. As shown in the table, for thefour-partition case, the coarsening heuristic and the sparsest cut heuristic provide solutions with

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Table 1. Comparison between the heuristic-based methodologies and the optimal solution. (Boldvalues in the table refers to the 7200 sec (2 hours) running time).

Coa

rsen

ing

Heu

rist

ic

No. of Partitions:Intra flow %:Inter flow %:Node Balancea:Flow Balanceb:Running Time (Sec):

291.408.60

00.130.017

484.9515.050.090.210.02

582.2717.730.360.580.02

870.7229.270.470.700.02

1066.6933.310.500.900.02

Spar

sest

Cut

Heu

rist

ic

No. of Partitions:Intra flow %:Inter flow %:Node Balance:Flow Balance:Running Time (Sec):

295.384.62

00.154.35

483.8316.17

00.2811.20

581.2618.740.070.1816.72

869.8630.14

00.6617.08

1066.5933.410.240.7223.78

Opt

imal

Sol

utio

n

No. of Partitions:Intra flow %:Inter flow %:Node Balance:Flow Balance:Running Time (Sec):

295.384.62

00.152.50

488.0211.98

00.19

614.70

585.0314.970.200.317200

869.8530.15

00.647200

1072.02 27.980.450.997200

aNode balance is measured in terms of CoV of number of nodes in the different network componentsbFlow balance is measured in terms of CoV of the link flows in the different network components

intra-flow percentages of 84.95% and 83.83%, respectively. The corresponding optimal intra-flow percentage is recorded to be 88.02%. While the sparsest cut heuristic provides a perfect nodebalance (CoV=0) for that case, the coarsening heuristic records a CoV of 0.09. As shown in thetable, the heuristics recorded significantly shorter execution time compared with that requiredfor obtaining the optimal solution. For the four-partition case, the coarsening and sparsest cutheuristics recorded a running time of 0.02 and 11.20 s, respectively. This difference in the runningtimes is due to the multiple activation of the MCF problem required by the sparsest cut heuristic.An execution time of 614.70 s is recorded to obtain the optimal solution for the same case. Itshould be noted that the mathematical program failed to provide the optimal solution for highernumber of partitions for this network. The results given in the table for the 5, 8, and 10-partitioncases are recorded after a 2 h running time.

Table 2 illustrates the network partitioning results using the two heuristics for a hypotheticalnetwork with radial topology and symmetric flow patterns. Two different link flow patterns areconsidered. The first pattern represents the case in which the dominant flows are along the radial

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Table 2. Network partitioning for a radial network with two different flow patterns.

Radial flow patternC

oars

enin

g h

euri

stic

No. of partitions:Intra flow %:Inter flow %: Node balance:Flow balance:

Running time (S):

287.512.5

00

0.01

39010

0.080.020.02

486.2513.750.210.320.02

582.517.50.230.47

0.013

8901000

0.03

16802000

0.02

Spar

sest

cut

heu

rist

ic

No. of partitions:Intra flow %:Inter flow %: Node balance:Flow balance:

Running time (S):

297.52.500

17.25

39010

0.030.04

22.82

495500

20.88

586.2513.75

00.08

25.22

8901000

28.25

16802000

31.13

Circular flow pattern

Coa

rsen

ing

heur

istic

No. of partitions:Intra flow %:Inter flow %: Node balance:Flow balance:

Running time (S):

294.665.34

00

0.02

389.3110.690.120.130.02

588.2111.79

00

0.02

1077.3522.65

00

0.02

1570.2329.770.820.900.04

2068.7

31.3000

0.04

Spar

sest

cut

heu

rist

ic

No. of partitions:Intra flow %:Inter flow %: Node balance:Flow balance:

Running time (S):

294.665.34

00

22.16

387.6012.400.030.06

46.11

588.2111.79

00

25.78

1077.3522.65

00

107.17

1567.1832.820.340.59

221.40

2068.7031.30

00

110.71

corridors in the network, while the second pattern considers a case in which the dominant flowsare along the network’s ring corridors. In both cases, the dominant flow value is assumed to befive times that of the flow value assigned to all other links. As shown in the table, the sparsest cutheuristic provides the optimal partitioning pattern for all cases where the network is symmetricallypartitioned (e.g. 2, 4, 8, and 16 partitions). For the coarsening heuristic, centroids are selectedmanually considering the network’s symmetric flow pattern. For example, in the radial flowpattern with two partitions case, the centroids are selected on two opposite radial corridors. Asillustrated, the coarsening heuristic is able to provide the optimal solution for some of the testedcases. It obtains the optimal partitions for the circular flow pattern with 2, 5, 10, and 20 partitions.However, it fails to provide the optimal solution for some other tested cases. For example, inthe case of radial flow pattern with two partitions, while both heuristics achieve perfect balance,the total intra-flow of the coarsening heuristic is recorded to be 87.5% compared with 97.5%that is obtained by the sparsest cut heuristic. The difference in the two solutions is explainedby their underlying approaches. The sparsest cut heuristic primarily searches for the minimumcuts in the network, while the coarsening heuristic searches for components with the maximum

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flows. In the two-partition case, the cut obtained by the coarsening heuristic is not the minimumcut as obtained by the sparsest cut heuristic. As for the running time, the coarsening heuristicsignificantly outperforms the sparsest cut heuristic in all tested cases. For instance, in the circularflow with 20 partitions, an execution time of 0.04 and 110.71 s is recorded for the coarseningheuristic and sparsest cut heuristic, respectively.

As a real-world application, the methodology is applied to provide an efficient partitioningfor the Dallas metroplex. The network is extracted from the Dallas-Fort Worth’s regional traveldemand model (North Central Texas Council of Governments, Transportation Department 2009),

Table 3. Network partitioning of Dallas network.

No. of partitions: 20Intra flow %: 91.45Inter flow %: 8.55Node balance: 0.47Flow balance: 0.76Running time (S): 4.07

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which consists of 5701 nodes and 9404 links. The link flows are taken as the sum of the morningand evening rush periods as estimated by the model. Considering the large size of this network,only the network coarsening heuristic was able to successfully execute and provide its solution.Tested on a more powerful server, the sparsest cut heuristic failed to converge at a reasonablerunning time. As mentioned earlier, the bottleneck is the step of determining the optimal MCFand its corresponding cut at each iteration. Table 3 illustrates the partitioning results for theDallas network considering 20 partitions. As described above, the heuristic starts by determiningthe set of centroids. For this experiment, nodes are first sorted following a descending order basedon the sum of their link flows. Nodes in this sorted list are then sequentially scanned. A nodeis selected to be a centroid only if it is separated from all previously selected centroids by apredefined distance that ensures that the selected centroids are well distributed in the network.The selected centroid nodes are illustrated in the figures by bold circles. Table 3 shows snapshotsof the interface of the implemented program while rendering the obtained subnetworks. The tableillustrates snapshots ranging from showing the first two obtained subnetworks till all subnetworksare produced. As shown in the snapshots, guided by the network’s estimated flow pattern, theresulted subnetworks take the shape of longitudinal corridors as well as bounded subareas. Thetotal intra-flow percentage is recorded to be more than 91% implying that the heuristic reasonablymaintains flow-independency among the subnetworks. A node balance CoV of 0.47 is recordedimplying that the resulting subnetworks are not of the exact same size. As described above, thecoarsening heuristic adopts the heavy link match strategy. Thus, it does not prohibit the occurrenceof the case where one subnetwork is bounded by its adjacent subnetworks with no opportunity tocombine more nodes, while the coarsening process is continuing for other subnetworks. Finally, arunning time of 4.07 s is recorded to obtain the illustrated partitioning, which is again an advantageof the network coarsening heuristic.

Finally, Table 4 illustrates the results of a set of experiments that examine the sensitivity of theobtained partitions for the Dallas network with respect to changes in the network’s link trafficflows. The goal of the experiment is to examine the need to modify the partitions due to changesin the traffic volumes at different parts of the network. The traffic flows of randomly selected 1%,5%, 10%, 25%, 50% and 100% of the links are assumed to increase by 1%, 5%, 10%, 15%, 20%and 100% of their initial value. The node-balance and flow-balance CoVs among the obtainedpartitions are recorded for all cases. Comparing these CoV measures with the base case in Table 3provides an indication on how the solutions are different. As shown in the table, a modification inthe network configuration due to changes either in the link flows or number of links with modified

Table 4. Sensitivity analysis of the coarsening heuristic.

% number of links with modified flow

% change in link flow 1 5 10 25 50 100

1 Node balance: 0.46 0.47 0.47 0.54 0.57 0.47Flow balance: 0.76 0.76 0.75 0.76 0.65 0.76

5 Node balance: 0.43 0.47 0.47 0.49 0.40 0.47Flow balance: 0.79 0.77 0.77 0.73 0.73 0.76

10 Node balance: 0.46 0.34 0.34 0.47 0.26 0.47Flow Balance: 0.77 0.92 0.83 0.64 0.84 0.76

15 Node Balance: 0.47 0.42 0.38 0.40 0.62 0.47Flow Balance: 0.76 0.86 0.87 0.81 0.62 0.76

20 Node Balance: 0.47 0.54 0.61 0.28 0.45 0.47Flow balance: 0.76 0.66 0.60 1.01 0.82 0.76

100 Node balance: 0.42 0.73 0.52 0.51 0.50 0.47Flow balance: 0.89 0.66 0.66 0.75 0.81 0.76

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flows could result in a different partitioning pattern. For instance, in the case where the volumesof 5% of the links in the network are increased by 100%, the node-balance CoV is recorded to be0.73 compared with 0.47 in the base case. Similarly, for the case in which the flows of 50% of thelinks are modified by 10%, the node-balance CoV is recorded to be 0.28. Both cases imply that thepartitioning patterns are significantly different from that obtained in the base case. However, forthe cases where the traffic flows of limited number of links in the network are slightly changed, thesolution is very close to that obtained in Table 3. For instance, when the flows of 1% of the linksare increased by 1% of their initial value, the node-balance CoV is almost the same as in Table 3.Similarly, as expected, for the cases in which the flows of all links in the networks are uniformlymodified (i.e. the last column in Table 4), the obtained partitions are shown to be identical to thosereported in the base case.

Summary

This paper aims to address a problem in large-scale traffic network management. It presents amethodology for partitioning urban transportation networks for distributed traffic managementapplications. The proposed methodology partitions the network such that: (a) the inter-flow amongthe resulting network components is minimised, (b) these network components are balanced interms of their sizes and/or traffic flows within their boundaries, and (c) each network componentis connected. A mathematical formulation of the problem is presented. Since the problem has beenshown to be NP-hard, two heuristics are developed. The first heuristic adopts a recursive approachin which the network is partitioned along its sparsest cuts till the balance condition is achieved.The second heuristic adopts a greedy-based network coarsening methodology. In this heuristic, thecoarsening process is iteratively applied along high-flow links, and terminates when the numberof nodes in the coarsened network is equal to the required number of partitions. The heuristicsare applied to obtain the partitioning pattern for two hypothetical networks. The results show thatboth heuristics provide near-optimal solution quality. In addition, the partitions suggest that linksthat are parts of a corridor with dominant flow remain in the same partition, as emphasised bythe results of the radial network. The running time of the sparsest cut increases significantly asthe network size increases. Determining the optimal MCF is the bottleneck that slows down theexecution of this heuristic. The network coarsening heuristic is used to determine an efficientpartitioning for the Dallas metroplex. The solution aims at maintaining each dominating trafficmovement within the network in one partition. The obtained network components are shownto cover longitudinal corridors as well as bounded subareas. The cut sections would generallyrepresent locations where these main traffic movements discontinue.

Several extensions are considered for this research work. For example, we will examine theeffect of adopting the presented network partitioning methodology on the performance of trafficmanagement schemes using a distributed architecture. We will compare the overall network per-formance when the network partitioning is made arbitrarily versus the case in which the networkis partitioned such that traffic inter-flow among the distributed controllers is minimised. In addi-tion, as explained above, the methodology adopts the MCF problem to provide an approximatesolution to the sparsest cut problem. Although the MCF problem is polynomially bounded, itis computationally intensive especially for large networks. Adopting heuristic-based approachesto determine a good approximation of the network’s sparsest cut is another extension. We willalso consider implementing a heuristic approach that combines the coarsening and sparsest cuttechniques. Following this approach, the network is coarsened such that the MCF can be solvedin a reasonable running time. The trade-off between the level of coarsening and the quality ofthe solution will be investigated. Finally, comparison between the performance of the presentedapproaches and other approaches in the literature will be investigated as an extension of this work.

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