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A Simple Efficient Approximation Scheme for the Restricted Shortest Path Problem

Dean H. Lorenz, Danny RazOperations Research Letter,

Vol. 28, No. 5, pages 213-219, June 2001

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Outline

Contribution and motivation Problem definition and Hassin’s results SPPP algorithm Improved Hassin’s algorithm SEA algorithm Conclusion

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Contribution

Propose a FPAS for RSP problem– Complexity of - approximation scheme

• • Valid for general graph with any cost values

A simple way to compute upper and lower bounds for RSP problem

A new test procedure

))/1log(log|(| nnEO

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Motivation

Based on Hassin’s original result with two improvements– achieve time complexity

• • applied to general graphs with any cost values

– How to find upper and lower bound such that

•

– Combine them to obtain claimed result

))/1)/log((log|(| LBUBnEO

nLBUB /

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Outline

Contribution and motivation Problem definition and Hassin’s results SPPP algorithm Improved Hassin’s algorithm SEA algorithm Conclusion

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RSP Problem Definition

Given – G(V,E) with |V|=n and |E|=m– Each edge is associated with

• Length (or cost) Ce

• Transition time (or delay) de

– Source and targets– A positive integer T

Ee

Vts ,

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Problem Definition (cont.)

Find– A path p in G from s to t satisfying

• Transition time (or delay) along the path is no greater than T

• Length (or cost) of path p is munimum

The problem is NP-complete, but has a FPAS– Path with cost no greater than

• c* is optimal cost

*)1( c

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Hassin’s Results

Given– An instance of RSP problem– Upper and lower bound of optimal value

• UB: sum of the n-1 longest edges• LB: 1

– Approximation factor An -approximated scheme with

–

))/log(log)/(( LBUBmnO

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Outline

Contribution and motivation Problem definition and Hassin’s results SPPP algorithm Improved Hassin’s algorithm SEA algorithm Conclusion

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Scaled Pseudo Polynomial Plus (SPPP algorithm)

A modified test procedure Definition of - test procedure

– Given :• An instance of RSP problem• Approximation factor and a value B

– Properties :• If answers YES, then • If answers NO, then

),( BT

),( BT BOPT ),( BT )1( BOPT

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SPPP Algorithm (cont.)

Idea – First scale cost values– Then run pseudo-polynomial algorithm to

find smallest delay for each cost Notation

– D(v,i) means minimum delay on a path from s to v with cost no more than i

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SPPP Algorithm (cont.)

Lemma 1:– Let p be any path, then the cost of p

satisfies

Proof :– , hence–

nSpcSpcpc )()(~)(

1/~/ SccSc lll ScScc lll ~

nSpcSpccScpcpl pl

ll

)()(~~)(

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SPPP Algorithm (cont.)

Lemma 2:– Any path p returned by SPPP satisfies

Proof :– – –

LUSnUpcc )1()(*

)(* pcc

Upc~

)(~

LUSnUSUSpc )1(~

)(~

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SPPP Algorithm (cont.)

Lemma 3:– If , then SPPP returns a feasible path

p that satisfies Proof :

– – – by lemma 1,–

Lcpc *)(

*cU

1/~ Scc ll

UnSUPSccpcpl

l

~/||/~)(~ **

*

LcnScSpc ***)(~

LcSpcSpcpc **)(~)(~)(

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Complexity

Overall complexity–

If , and –

)()~

( nL

UnmOUmO

)()~

(L

UnmOUmO

LU 1

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SPPP Algorithm (cont.)

Lemma 4:– If returns

FAIL, then test T(1,B)=Yes, otherwise T(1,B)=No

– T(1,B) is a 1-test Complexity

– Call SPPP with U=L=B and requires O(mn)

)1,,,,},{),,(( BBTcdEVGSPPP Elll

1

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Outline

Contribution and motivation Problem definition and Hassin’s results SPPP algorithm Improved Hassin’s algorithm SEA algorithm Conclusion

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Improved Hassin’s Algorithm (Hassin’) Idea

– Initial bound BL=LB, BU=

– If , 2BU is a valid upper bound

– Then use bounds with algorithm SPPP Theorem

– Given valid bounds ; an -approximate solution can be found in

2/ LU BB 2/UB

UB c LB *

0

)loglog(LB

UBmn

mnO

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Hassin’ Algorithm

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Complexity

Complexity of finding bounds– Binary search requires tests– Each test requires steps– Find B in :

Complexity of call SPPP–

)log(logLB

UBO

)(mnO

)loglog(LB

UBmnO

)(mn

O

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Outline

Contribution and motivation Problem definition and Hassin’s results SPPP algorithm Improved Hassin’s algorithm SEA algorithm Conclusion

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Simple Efficient Approximation (SEA) Algorithm

Objective– Find upper and lower bound for optimal

value such that ratio between them is n Notation

– be distinct edge length – – , for – , and for

lccc ...21

mEl ||

}|{ iei ccEeE li 1

GGEVG lii ),,( 1 ii GG 10 li

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SEA Algorithm

Idea– must have a T-path – Exist a unique index j

• has a T-path• does not have a T-path• then

lG)1( lj

jG

1jG

jj ncOPTcc *

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SEA Algorithm (cont.)

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Complexity

Theorem:– Algorithm SEA is a FPAS for RSP problem

with complexity Complexity

– times complexity of shortest path algorithm

– Second part is :– Dominant is second part

))/1log(log( nmnO

)(logmO

))/1log(log( nmnO

)log( nnmO

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Conclusion

Main contribution– Improve complexity– Enlarge scope of FPAS for RSP problem

Future work– Can be applied to problems with similar

characteristics• QoS routing and partition