a grasp with path-relinking heuristic for pvc routing

February 2002

GRASP with path-relinking for PVC routingSlide 1/42 (ROADEF)

A GRASP with path-relinking heuristic for

PVC routingCelso C. Ribeiro

Computer Science Department

Catholic University of Rio de Janeiro

joint work with M.G.C. Resende

February 2002


Summary

• PVC routing• Integer multicommodity flow

formulation• Cost function• Solution method: GRASP with path-

relinking• Numerical results• Conclusions

February 2002


PVC routing: application

• Virtual private networks: permanent virtual circuits (PVCs) between customer endpoints on a backbone network

• Routing: either automatically by switch or by network designer without any knowledge of future requests

• Inefficiencies and occasional need for off-line rerouting of the PVCs

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PVC routing: example

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PVC routing: examplemax capacity = 3

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PVC routing: examplemax capacity = 3very long path!

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PVC routing: examplemax capacity = 3very long path!

reroute

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PVC routing: examplemax capacity = 3

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PVC routing: examplemax capacity = 3feasible and

optimal!

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• Other algorithms simply handle the number of hops (e.g. routing algorithm in Cisco switches)

• Handling delays is particularly important in international networks, where distances between backbone nodes vary considerably

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• Reorder PVCs and apply algorithm on switch to reroute: – taking advantage of factors not

considered by switch algorithm may lead to greater network efficiency

– FR switch algorithm is typically fast since it is also used to reroute in case of switch or trunk failures

– this can be traded off for improved network resource utilization when routing off-line

February 2002


PVC routing: application• Load balancing is important for

providing flexibility to handle:– overbooking: typically used by network

designers to account for non-coincidence of traffic

– PVC rerouting: due to failures– bursting above the committed rate: not

only allowed, but also sold to customers as one of the attractive features of frame relay

• Integer multicommodity network flow problem

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Problem formulation

• Given undirected FR network G = (V, E), where– V denotes n backbone nodes (FR switches)– E denotes m trunks connecting backbone

nodes• for each trunk e = (i,j )

– b (e ): maximum bandwidth (max kbits/sec rate)

– c (e ): maximum number of PVCs that can be routed on it

– d (e ): propagation and hopping delay

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Problem formulation

• Demands K = {1,…,p } defined by– Origin-destination pairs (o,d )– r (p): effective bandwidth requirement

(forward, backward, overbooking) for PVC p

• Objective is to minimize– delays– network load unbalance

• subject to– technological constraints

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Problem formulation

• route for PVC (o,d ) is a sequence of adjacent trunks from node o to node d

• set of routing assignments is feasible if for all trunks e– total bandwidth requirements routed on

e does exceed b (e)– number of PVCs routed on e not greater

than c(e)

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Problem formulation

, ,( , ) ( , )

1

, , ,

1, , , , ,

,

, , ,

( ) ,

1,

min ( ) ( ,..., , ,..., )

subject

if is source for

1

to

( ) , (

, if

, ) ,

( ) , ( , ) ,k kk i j j i i j

k

k ki j j i i j

k K

k ki j

p ki j i j i j j i j i

i j E i j

j i

K

i j E i j E

x x x

r

i V k

x x b i j E

x x c i j E i j

x

j

x

i

x

K

x

,

is destination for

0, other

0,1 , ( , ) ,

wise

.ki jx i j

i K

E k K

V k

,ki jx= 1, iff trunk (i,j )

is used to route PVC k.

,ki jx

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Cost function

• Linear combination of – delay component - weighted by (1-)– load balancing component - weighted

by

• Delay component: , , ,( )k ki j k i j j ik K

d x x

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Cost function

• Load balancing component: measure of Fortz & Thorup (2000) to compute congestion:

= 1(L1) + 2(L2) + … + |E|(L|

E|)

where Le is the load on link e E,

e(Le) is piecewise linear and convex,

e(0) = 0, for all e E.

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Piecewise linear and convex e(Le) link

congestion measure

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1 1.2

cost

per

unit

of ca

paci

ty

trunk utilization rate

slope = 1slope = 3 slope = 10

slope = 70

slope = 500

slope = 5000

(Lece)

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Some recent applications• Laguna & Glover (1993): tabu search, different

cost function, no constraints on PVCs routed on the same trunk (assign calls to paths)

• Sung & Park (1995): Lagrangean heuristic, very small graphs

• Amiri et al. (1999): Lagrangean heuristic, min delay

• Dahl et al. (1999): cutting planes (traffic assignment)

• Barnhart et al (2000): branch-and-price, different cost function, no constraints on PVCs routed on same trunk

• Shyur & Wen (2000): tabu search, min hubs

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Solution method: GRASP with

path-relinking• GRASP: Multistart metaheuristic, Feo & Resende

1989• Path-relinking: intensification, Glover (1996)• Repeat for Max_Iterations:

– Construct a greedy randomized solution– Use local search to improve the constructed

solution– Apply path-relinking to further improve this solution– Update the pool of elite solutions– Update the best solution found

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Solution method: GRASP

• GRASP– Construction:

• RCL: nc unrouted PVCs with largest demands• choose unrouted pair k biasing in favor of high

bandwidth requirements, with probablity k = rk / (pRCL rp )

• capacity constraints relaxed and handled via the penalty function introduced by the load-balancing component

• length of each edge (i,j) is the incremental cost of routing rk additional units of demand on it

• route pair k using shortest route between its endpoints

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Solution method: GRASP

• GRASP– Local search:

• for each PVC k K , remove rk units of flow from each edge in its current route

• recompute incremental weights of routing rk additional units of flow for all edges

• reroute PVC k using new shortest path

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Solution method: path-relinking

• Introduced in the context of tabu search by Glover (1996)– Intensification strategy using set of

elite solutions

• Consists in exploring trajectories that connect high quality solutions.

initialsolution

guidingsolution

path in neighborhood of solutions

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• Path is generated by selecting moves that introduce in the initial solution attributes of the guiding solution.

• At each step, all moves that incorporate attributes of the guiding solution are evaluated and the best move is taken:

Initialsolution

guiding solution

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Elite solutions x and y(x,y): symmetric difference

between x and ywhile ( |(x,y)| > 0 ) {

evaluate moves corresponding in (x,y) make best move

update (x,y)}


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Path-relinking in GRASP

• Maintain an elite set of solutions found during GRASP iterations.

• After each GRASP iteration (construction & local search):– Select an elite solution at random:

guiding solution.– Use GRASP solution as initial solution.– Perform path-relinking between these

two solutions.

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Path-relinking in GRASP

• Successful applications:– Prize-collecting Steiner tree problem

Canuto, Resende, & Ribeiro (2000)– Steiner tree problem

Ribeiro, Uchoa, & Werneck (2000) (e.g., best known results for open problems in series dv640 of the SteinLib)

– Three-index assignment problem Aiex, Pardalos, Resende, & Toraldo (2000)

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Path-relinking: elite set

• P is set of elite solutions• Each iteration of first |P | GRASP

iterations adds one solution to P (if different from others).

• After that: solution x is promoted to P if:– x is better than best solution in P.– x is not better than best solution in P, but is

better than worst and is sufficiently different from all solutions in P .

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Experiment

• Heuristics:– H1: sorts demands in decreasing order and

routes them using minimum hops paths– H2: sorts demands in decreasing order and

routes using same cost function as GRASP– H3: adds the same local search to H2– GPRb: GRASP with backwards path-relinking

• SGI Challenge 196 MHz

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Experiment• Test problems:

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• Variants of path-relinking:– G: pure GRASP– GPRb: GRASP with backward PR– GPRf: GRASP with forward PR– GPRbf: GRASP with two-way PR

• Other strategies:– Truncated path-relinking– Do not apply PR at every iteration

(frequency)

Variants of GRASP and path-relinking

S T

TS

S T

S T

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10 100 1000 10000 100000 1e+06

GGPRfGPRb

GPRfb

Time (s)

Pro

bab

ility

Each variant: 200 runs for one instance of PVC routing problem

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• Same computation time: probability of finding a solution at least as good as the target value increases from G GPRf GPRfb GPRb

• P(h,t) = probability variant h finds solution as good as target value in time no greater than t– P(GPRfb,100s)=9.25% P(GPRb,100s)=28.75%– P(G,2000s)=8.33% P(GPRf,2000s)=65.25%

• P(h,time)=50% Times for each variant: – GPRb:129s G:10933s GPRf:1727s

GPRfb:172s

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Comparisons

Distribution: 86/60/2: 86 edges with utilization in [0,1/3), 60 in [1/3,2/3), and two in [2/3,9/10)

In general: GPRB > H3 > H2 > H1 (cost, max utilization, distribution)

costmax util.

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Parameter of the objective function• Objective function (solution) = Delay x (1-) +

Load imbalance cost x

• if = 1: consider only trunk utilization rates• if = 0: consider only delays (capacities relaxed)• increasing 0 1 minimization of maximum

utilization rate dominates reduction of flows in edges with higher loads increase of flows in underloaded edges until the next breakpoint flows concentrate around breakpoint levels useful strategy for setting appropriate value of to achieve some level of quality of service (max util.)

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Parameter of the objective function

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0.0001 0.001 0.01 0.1 192000

94000

96000

98000

100000

102000

104000

maxim

um

utiliz

ation

dela

y

delta

delay

maximum utilization

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Concluding remarks (1/3)

• New formulation with flexible objective function

• Family of heuristics (greedy, greedy+LS, GRASP, GRASP+PR)

• Simple greedy heuristic improves algorithm in traffic engineering by network planners

• Objective function provides effective strategy for setting the weight parameter to achieve some quality of service level

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Concluding remarks (2/3)

• Path-relinking adds memory and intensification mechanisms to GRASP, systematically contributing to improve solution quality.

• Some implementation strategies appear to be more effective than others (e.g., backwards from better, elite solution to current locally optimal solution).

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Concluding remarks (3/3)• NETROUTER – Tool for optimally loading

demands on single-path routes on a capacitated network. It uses the GPRb variant of the combination of GRASP and path-relinking, minimizing delays while balancing network load.

• Application - Netrouter is currently being used for the design of AT&T's next generation frame-relay and MPLS core architecture, to assess if the current and forecasted demands can be handled by the proposed trunking plan.

February 2002


Slides and publications

• Slides of this talk can be downloaded from: http://www.inf.puc-rio/~celso/talks

• Paper about PVC routing available at:

http://www.inf.puc-rio.br/~celso/publicacoes

a grasp with path-relinking heuristic for pvc routing

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