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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.

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Lecture 2-3 Bharathi-Kempe-Salek Conjecture

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  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!

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