1 / 30

WALCOM 2012

On the Hardness of Point-Set Embeddabiltiy. Stephane Durocher Debajyoti Mondal. Department of Computer Science University of Manitoba. WALCOM 2012. February 16, 2012. Point-Set Embeddings. b. c. h. a. i. g. f. e. d. A point set P. A plane graph G.

jerica
Télécharger la présentation

WALCOM 2012

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 the Hardness of Point-Set Embeddabiltiy Stephane Durocher DebajyotiMondal Department of Computer Science University of Manitoba WALCOM 2012 February 16, 2012

  2. Point-Set Embeddings b c h a i g f e d A point set P A plane graph G WALCOM 2012 February 16, 2012 1

  3. Point-Set Embeddings b b c c h h a a i i g g f f e e d d An embedding of G on P A plane graph G WALCOM 2012 February 16, 2012 2

  4. Previous Results Reference Graph Class Time complexity WALCOM 2012 February 16, 2012 3

  5. Sketch of Cabello’s Proof S c1 c1 9 c2 10 c|S| 11 3-Partition  Point-Set Embeddability (2-connected graphs) S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32 x x y y A chain S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11} WALCOM 2012 February 16, 2012 4

  6. Sketch of Cabello’s Proof S c1 9 c2 10 c|S| 11 3-Partition  Point-Set Embeddability (2-connected graphs) y x S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32 x y S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11} WALCOM 2012 February 16, 2012 5

  7. Sketch of Cabello’s Proof S c1 9 c2 10 c|S| 11 3-Partition  Point-Set Embeddability (2-connected graphs) y x S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32 x y G S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11} WALCOM 2012 February 16, 2012 6

  8. Sketch of Cabello’s Proof B B B B 3-Partition  Point-Set Embeddability (2-connected graphs) y x S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32 G S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11} P WALCOM 2012 February 16, 2012 7

  9. Sketch of Cabello’s Proof y 3-Partition  Point-Set Embeddability (2-connected graphs) y x S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32 x G S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11} P WALCOM 2012 February 16, 2012 8

  10. Sketch of Cabello’s Proof y 3-Partition  Point-Set Embeddability (2-connected graphs) y x S = {9, 10, 14, 12, 10, 9, 12, 11, 9, 10, 11, 11 } , B = 32 x G S1={10, 10, 12} , S2={ 9, 11, 12} , S3={ 9, 9,14} , S4={ 10, 11,11} P WALCOM 2012 February 16, 2012 9

  11. Sketch of Kaufmann and Wiese’s Proof Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs) v3 v4 v6 v5 v1 v2 P G Does G contain a Hamiltonian Cycle? Does G admits a 1-bend PSE on P ? WALCOM 2012 February 16, 2012 10

  12. Sketch of Kaufmann and Wiese’s Proof Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs) v3 v3 v2 v5 v4 v6 v4 v6 v1 v6 v1 v2 v5 v4 v3 v5 v1 v2 P G If G contains a Hamiltonian Cycle, then G admits a 1-bend PSE on P WALCOM 2012 February 16, 2012 11

  13. Sketch of Kaufmann and Wiese’s Proof Hamiltonian Cycle  1-Bend P.S.E. (3-connected graphs) v3 How to get rid off bends? v4 e v6 v6 v1 v2 v5 v4 v3 v5 v1 v2 G P If G admits a 1-bend PSE on P, then G contains a Hamiltonian Cycle WALCOM 2012 February 16, 2012 12

  14. Our Result Point-Set Embeddability is NP-hard for 3-Connected Graphs G G / WALCOM 2012 February 16, 2012 13

  15. Lets Try... v3 Does G contain a Hamiltonian Cycle? v4 v6 ? ? v5 ? v1 v2 G Does G /admits a PSE on P ? P G / WALCOM 2012 February 16, 2012 14

  16. Problems… ? P P G / WALCOM 2012 February 16, 2012 15

  17. Tricks… c c a c a a G d d d b h h b h g b g g f f f e e e G / WALCOM 2012 February 16, 2012 16

  18. Idea of Reduction A c a d h b g f G e c a d b h g P f e G / B WALCOM 2012 17

  19. Idea of Reduction A c a d h b g f G e g f d c b h e a c a d b h g P f e G / B WALCOM 2012 18

  20. Idea of Reduction A A g f d c b h e a B B G / WALCOM 2012 19

  21. Multiple Point-Sets A 12 12 points 12 c c a a 14 d d b b h h g g f f 12 points e e 10 G / B G / WALCOM 2012 20

  22. PSE is NP-hard for 3-Connected Graphs c a Does G contain a Hamiltonian Cycle? d h b g f G e c a Does G / admit a PSE on some point set among P1, P2, … , Pk? d b h g f e G / WALCOM 2012 February 16, 2012 21

  23. Positive Results b Nishat et al. (2010), Durocher et al. (2011), Moosa and Rahman (2011) Point-set embeddability can be tested for plane 3-trees in O(n4/3 + ɛ) time. b k k k e d e e h d h a f f g g g c a m c m m c A plane 3-tree G A construction for G WALCOM 2012 February 16, 2012 22

  24. Graphs with Plane 3-Trees as Weak Dual c c c c i i g j j h h d l f f f k e e m a a a a b b b b WALCOM 2012 February 16, 2012 23

  25. Graphs with Plane 3-Trees as Weak Dual c c c c i i g j j h h d l f f f k e e m a a a a b b b b f f m f d e e h h l g k j j i i b b b b a a a a c c c c WALCOM 2012 February 16, 2012 24

  26. Convex PSE of Klee Graphs a q p n o m l k j f r d i c h e c b g r P A klee graph G m h l g n j i k p e d Does G admits a convex point-set embedding on P ? o f q a b WALCOM 2012 February 16, 2012 25

  27. Convex PSE of Klee Graphs a q p n o m l k j f r d i h e c b g P A klee graph G Does G admits a convex point-set embedding on P ? WALCOM 2012 February 16, 2012 26

  28. Dynamic Programming q p o n k m a l j f a r d i q e g q h p n p n o o k m l k m f l j r j d f r i d e g i h h c b e c b g WALCOM 2012 February 16, 2012 27

  29. Future Works • Is PSE NP-hard for 4-connected graphs? • Convex PSE algorithms for general kleegraphs . • PSE algorithms for klee graphs without convexity constraint. WALCOM 2012 February 16, 2012 28

  30. Thank You..

More Related