A Simple Efficient Approximation Scheme for the Restricted Shortest Path Problem Dean H. Lorenz, Danny Raz Operations Research Letter, Vol. 28, No. 5, pages 213-219, June 2001

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

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

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

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

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

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

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

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

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

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

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

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

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

Complexity Overall complexity If , and

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)

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

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

Hassin’ Algorithm

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

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

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

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

SEA Algorithm (cont.)

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

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