1 / 43

Lecture 2-3 Bharathi-Kempe-Salek Conjecture

Lecture 2-3 Bharathi-Kempe-Salek Conjecture. Ding-Zhu Du University of Texas at Dallas. Bharathi-Kempe-Salek Conjecture. Solution. Deterministic diffusion model - polynomial-time . Linear Threshold (LT) – polynomial-time . Independent Cascade (IC) – NP-hard.

psteele
Download Presentation

Lecture 2-3 Bharathi-Kempe-Salek Conjecture

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. Lecture 2-3 Bharathi-Kempe-Salek Conjecture • Ding-Zhu Du • University of Texas at Dallas

  2. Bharathi-Kempe-Salek Conjecture

  3. Solution • Deterministic diffusion model -polynomial-time. • Linear Threshold (LT) – polynomial-time. • Independent Cascade (IC) – NP-hard.

  4. Deterministic Diffusion Model • When a node becomes active (infected or protected), it activates all of its currently inactive (not infected and not protected) neighbors. • The activation attempts succeed with a probability 1.

  5. Deterministic Model 6 2 1 5 3 4 both 1 and 6 are source nodes. Step 1: 1--2,3; 6--2,4. .

  6. Example 6 2 1 5 3 4 Step 2: 4--5.

  7. A Property of Optimal Solution

  8. Naïve Dynamic Programming

  9. Naïve Dynamic Programming

  10. Running Time It is not a polynomial-time!

  11. Counting

  12. Virtual Nodes Change arborescence to binary arborescence At most n virtual nodes can be introduced.

  13. Weight

  14. Naïve Dynamic Programming

  15. Linear Threshold (LT) Model • A node v has random threshold ~ U[0,1] • A node v is influenced by each neighbor w according to a weight bw,v such that • A node v becomes active when at least (weighted) fraction of its neighbors are active

  16. Example Inactive Node Y 0.6 Active Node Threshold 0.2 0.2 0.3 Active neighbors X 0.1 0.4 U 0.3 0.5 Stop! 0.2 0.5 w v

  17. A property

  18. Equivalent Networks

  19. At seed v

  20. At non-seed v

  21. At non-seed v

  22. At non-seed v

  23. At non-seed v

  24. At seed v

  25. Independent Cascade (IC) Model • When node v becomes active, it has a single chance of activating each currently inactive neighbor w. • The activation attempt succeeds with probability pvw . • The deterministic model is a special case of IC model. In this case, pvw =1 for all (v,w).

  26. Example Y 0.6 Inactive Node 0.2 0.2 0.3 Active Node Newly active node U X 0.1 0.4 Successful attempt 0.5 0.3 0.2 Unsuccessful attempt 0.5 w v Stop!

  27. At non-seed v

  28. Another Dynamic Programming

  29. Proof of NP-hardness

  30. Partition Problem This is a well-known NP-complete problem!

  31. Special Case This is still an NP-complete problem!

  32. Subsum Problem This is still an NP-complete problem!

  33. Key Fact 1

  34. <1?

  35. h=?

  36. Key Fact 2

  37. References

  38. THANK YOU!

More Related