Lecture 2 3 BharathiKempeSalek Conjecture DingZhu Du University

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

Bharathi-Kempe-Salek Conjecture 2

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

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

Deterministic Model 2 6 1 5 3 4 both 1 and 6 are source nodes. Step 1: 1 --2, 3; 6 --2, 4. . 9/15/2021 5

Example 2 6 1 5 3 4 Step 2: 4 --5. 9/15/2021 6

A Property of Optimal Solution 7

Naïve Dynamic Programming 8

Naïve Dynamic Programming 9

Running Time It is not a polynomial-time! 10

Counting 11

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

Weight 13

Naïve Dynamic Programming 14

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

Example Inactive Node Y 0. 6 0. 3 Active Node 0. 2 X Threshold 0. 2 Active neighbors 0. 1 0. 4 U 0. 5 w 0. 3 0. 5 Stop! 0. 2 v

A property 17

Equivalent Networks 18

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At seed v 21

At non-seed v 22

At non-seed v 23

At non-seed v 24

At non-seed v 25

At seed v 26

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).

Example Y 0. 6 Inactive Node 0. 3 0. 2 X 0. 4 0. 5 w 0. 2 U 0. 1 0. 3 0. 2 0. 5 v Stop! Active Node Newly active node Successful attempt Unsuccessful attempt

At non-seed v 29

Another Dynamic Programming 30

Proof of NP-hardness

Partition Problem This is a well-known NP-complete problem! 32

Special Case This is still an NP-complete problem! 33

Subsum Problem This is still an NP-complete problem! 34

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Key Fact 1 36

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<1？ 38

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h=? 40

Key Fact 2 41

References 42

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