1. A Simple Efficient Approximation Scheme for the Restricted Shortest Path Problem
Dean H. Lorenz, Danny Raz Operations Research Letter, Vol. 28, No. 5, pages , June 2001

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

3. 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

4. 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

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

6. 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

7. 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

8. Hassin’s Results Given An -approximated scheme with
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

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

10. 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

11. SPPP Algorithm (cont.) Idea Notation 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

13. SPPP Algorithm (cont.) Lemma 1: Proof :
Let p be any path, then the cost of p satisfies
Proof :
, hence

14. SPPP Algorithm (cont.) Lemma 2: Proof :
Any path p returned by SPPP satisfies
Proof :

15. SPPP Algorithm (cont.) Lemma 3: Proof :
If , then SPPP returns a feasible path p that satisfies
Proof :
by lemma 1,

16. Complexity
Overall complexity
If , and

17. SPPP Algorithm (cont.) Lemma 4: Complexity
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)

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

19. 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

20. Hassin’ Algorithm

21. Complexity Complexity of finding bounds Complexity of call SPPP
Binary search requires tests
Each test requires steps
Find B in :
Complexity of call SPPP

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

23. 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

24. SEA Algorithm Idea must have a T-path Exist a unique index j
has a T-path
does not have a T-path
then

25. SEA Algorithm (cont.)

26. Complexity Theorem: Complexity
Algorithm SEA is a FPAS for RSP problem with complexity
Complexity
times complexity of shortest path algorithm
Second part is :
Dominant is second part

27. Conclusion Main contribution Future work Improve complexity
Enlarge scope of FPAS for RSP problem
Future work
Can be applied to problems with similar characteristics
QoS routing and partition