1 / 27

A Simple Efficient Approximation Scheme for the Restricted Shortest Path Problem

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

hina
Download Presentation

A Simple Efficient Approximation Scheme for the Restricted Shortest Path Problem

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

  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 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 • 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

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

  13. SPPP Algorithm (cont.) • Lemma2: • Any path p returned by SPPP satisfies • Proof :

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

  15. Complexity • Overall complexity • If , and

  16. SPPP Algorithm (cont.) • Lemma 4: • If returns FAIL, then testT(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)

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

  18. 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

  19. Hassin’ Algorithm

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

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

  22. 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

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

  24. SEA Algorithm (cont.)

  25. 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

  26. 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

More Related