1 / 27

On Approximating the Maximum Simple Sharing Problem

On Approximating the Maximum Simple Sharing Problem. Danny Chen University of Notre Dame Rudolf Fleischer, Jian Li , Zhiyi Xie, Hong Zhu Fudan University. Restricted NDCE Problem. NDCE = N ode- D uplication based C rossing E limination

dolan
Download Presentation

On Approximating the Maximum Simple Sharing 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. On Approximating the Maximum Simple Sharing Problem Danny Chen University of Notre Dame Rudolf Fleischer, Jian Li, Zhiyi Xie, Hong Zhu Fudan University

  2. Restricted NDCE Problem • NDCE = Node-Duplication based Crossing Elimination • Design of circuits for molecular quantum-dot cellular automata (QCA)

  3. Restricted NDCE Problem • Duplicate and rearrange upper nodes • Each duplicated node can connect to only one node in V • Maintain all connections U 1 2 3 4 information V a b c d e

  4. Restricted NDCE Problem Naive method: duplicate |E|-|U| nodes U 2’ 1 1’ 3 2 2’’ 1’’ 4’ 3’ 4’ 4 3’’ 2’’’ 9 V c a b d e

  5. Restricted NDCE Problem Duplicated nodes can connect to only one node in V U 2’ 1 3 2 1’’ 4’ 3’ 4 3’’ 2’’’ 6 V c a b d e

  6. Maximum Simple Sharing Problem U 1 2 3 4 3 sharings V a b c d e

  7. Restricted NDCE Problem Duplicated nodes can connect to only one node in V U 2’ 1 4’ 2 3’ 4 3 1’ 2’’ 5 V b a c e d

  8. Maximum Simple Sharing Problem U 1 2 3 4 4 sharings V a b c d e

  9. U U 1 1 2 2 3 3 4 4 V V a a b b c c d d e e Maximum Simple Sharing Problem Goal: • Find simple node- disjoint paths • Start/end points in V • Maximize number of covered U-nodes

  10. duplicate |E| − |U| − m nodes of U m simple sharings Minimize #duplications is equivalent to maximize #simple sharings

  11. Cyclic Maximum Simple Sharing Problem (CMSS) Allow cycles! 1 2 3 4 a b c d e

  12. CMSS Reduction to maximum weight perfect matching problem 0 1

  13. CMSS Reduction to maximum weight perfect matching problem

  14. CMSS Reduction to maximum weight perfect matching problem

  15. CMSS Reduction to maximum weight perfect matching problem

  16. CMSS Reduction to maximum weight perfect matching problem

  17. CMSS max number of sharings = max weight of perfect matching

  18. From CMSS to MSS Arbitrarily breaking cycles gives a 2-approximation 1 2 3 4 a b c d e

  19. From CMSS to MSS Arbitrarily breaking cycles gives a 2-approximation 1 2 3 4 OPT=4 SOL=2 a b c d e

  20. 5/3-Approximation • Start with optimal CMSS solution • Do transformations, if possible • Done after polynomial number of steps

  21. Summary • 5/3-approximation of MSS by solving CMSS optimally and then breaking cycles in a clever way • Bound is tight for our algorithm • We have also studied the Maximum Sharing Problem (sharings can overlap)

  22. THANKS~

  23. U: 1 2 3 4 V: a b c d e Maximum Simple Sharing Problem 3 Sharings

  24. CMSS CMSS can be solved in polynomial time (reduction to a maximum weight perfect matching problem)

  25. From CMSS to MSS Improve the approximation ratio to 5/3

  26. From CMSS to MSS Improve the approximation ratio to 5/3 Cycle-breaking Algorithm • From the optimal solution of CMSS problem. • Repeatly do the 3 operations until no one applies. • Each operation can be implement in poly time. • We can show the algorithm terminate with poly steps.

More Related