Transcript On the target set selection problem

```On the target set selection
problem

Potential Customers
2
Free Samples
3
Word-of-mouth
4
Marketing
5
Marketing
6
Contagion
7
A Social Network with Threshold Function
(G, )
2
1
4
1
3
3
1
2
2
2
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Target Set S
2
1
4
1
3
3
1
2
2
2
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Activation Process Starting form S
G
[S ]
2
1
4
1
3
3
1
2
2
2
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min-seed (G,θ)
2
1
4
1
3
3
1
2
2
2
min-seed(G, )  1
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TARGET SET SELECTION
TARGET SET SELECTION:
Finding a target set S of smallest possible size that influences
all vertices in (G, ), that is [ S ]G  V (G).
Such an S is called a minimum seed or an optimal target set for (G, ).
min-seed(G, )   S : S  V (G) and [ S ]G  V (G)
13
Threshold Models
Constant threshold:
 (v)  k for all vertices v in G.
Majority threshold  :
 d (v ) 
for all vertices v in G.

 2 
  (v )  
 d (v )  1 
for all vertices v in G.

 2 
Strict majority threshold  :  (v)  
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Updating rules
Parallel updating rule:
All white vertices v that have at least  (v) black neighbors at the previous
round are colored black. The colors of the other vertices do not change.
Sequential updating rule:
Exactly one of white vertices that have at least  (v) black neighbors at
the previous round is colored black. The colors of the other vertices do
not change.
15
Parallel = Sequential
Lemma: Let (G, ) be a connected graph G with thresholds  on V(G).
An optimal target set for (G, ) under the sequential updating rule is
also an optimal target set for (G, ) under the parallel updating rule,
and vice versa.
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A Lower bound for min-seed (G,θ)
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[Peleg, 2002]
It is NP-hard to compute the optimal target set for majority thresholds.
[Dreyer and Roberts, 2009]
In constant threshold model, it is NP-hard to compute the min-seed (G, k )
for any k  3.
[Chen Ning, 2009]
The TARGET SET SELECTION problem is NP-hard when the thresholds
are at most 2.
19
[Chen Ning, 2009]
Given any regular graph with thresholds  (v)  2 ( or  (v)  2) for any
vertex v, the TARGET SET SELECTION problem can not be approximated
1
within the ratio of O(2log
n
) , for any fixed constant   0 , unless
NP  DTIME(n polylog ( n) ).
20
Results for Trees
[Dreyer and Roberts, 2009]
When G is a tree, the TARGET SET SELECTION problem can be solved in
linear time for constant thresholds.
[Chen Ning, 2009]
When the underlying graph is a tree, the problem can be solved in
polynomial-time under a general threshold model.
[Ben-Zwi et al, 2010]
For n-vertices graph G with treewidth bounded by  , the TARGET SET
SELECTION problem can be solved in nO ( ) time.
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Block-cactus graph
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vertex-sum at v of G1 and G2
G1 ⊕v G2
v
v
G1
v
G2
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(G1 ⊕v G2 , θ)
G1  v
v
1 ( x)   ( x) 1 for x  NG (v)
2 ( y)   ( y) for y V (G2  v)
1 ( x)   ( x) for x V (G1  v) \ NG (v)
 2 (v)   (v)  N G (v)  [ S1 ]G
S1 : an optimal target set for (G1  v,1 )
S2 : an optimal target set for (G2 ,2 )
1
1
1
1
that maximizes NG1 (v)  [ S1 ]G1
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(G  v,1 )
(G, )
(G  v,1 )
v
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Optimal Target Set
3
2
5
1
2
2
2
2
5
3
1
3
5
1
2
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
3
2
5
1
2
2
2
2
5
3
1
3
5
1
2
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
2
4c
2
b
4
a
3
2
5
2
2
2
5
d
5e
1
2
3
1
3
5
1
2
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
2
4c
2
b
4
a
3
2
5
2
5
3
1
5e
3
2 b
5
2
d
1
2
2
6
5
e
2
2
2
4
2
2
d
c
1
2
2
2
2
2
2
a4
1
1
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Optimal Target Set
2
4c
2
b
4
a
3
2
5
2
5
3
1
5e
3
2 b
5
2
d
1
2
2
6
5
e
2
2
2
4
2
2
d
c
1
2
2
2
2
2
2
a4
1
1
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Optimal Target Set
2
4c
2
b
4
a
3
2
5
2
5
3
1
5e
3
2 b
5
2
d
1
2
2
6
5
e
2
2
2
4
2
2
d
c
1
2
2
2
2
2
2
a4
1
1
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Optimal Target Set
2
4c
2
b
4
a
3
2
5
2
5
3
1
5e
3
2 b
5
2
d
1
2
2
6
5
e
2
2
2
4
2
2
d
c
1
2
2
2
2
2
2
a4
1
1
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Optimal Target Set
2
4c
2
b
4
a
3
2
5
2
5
3
1
5e
3
2 b
5
2
d
1
2
2
6
5
e
2
2
2
4
2
2
d
c
1
2
2
2
2
2
2
a4
1
1
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Optimal Target Set
2
4c
2
b
4
a
3
2
5
2
5
3
1
5e
3
2 b
5
2
d
1
2
2
6
5
e
2
2
2
4
2
2
d
c
1
2
2
2
2
2
2
a4
1
1
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Optimal Target Set
2
4c
2
b
4
a
3
2
5
2
5
3
1
5e
3
2 b
5
2
d
1
2
2
6
5
e
2
2
2
4
2
2
d
c
1
2
2
2
2
2
2
a4
1
1
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Optimal Target Set
2
4c
2
b
4
a
3
2
5
2
5
3
1
5e
3
2 b
5
2
d
1
2
2
6
5
e
2
2
2
4
2
2
d
c
1
2
2
2
2
2
2
a4
1
1
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Optimal Target Set
3
2
5
1
2
2
2
2
5
3
1
3
5-2
1
2
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
3
2
5
1
2
2
2
2
5
3
1
3
3
1
2
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
3
2
5
1
2
2
2
2
5
3
1
3
3
1
2
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
0 2 2 2 1 2 2 3 1 2 2 2 1
3
2
5
1
2
2
2
2
5
3
1
3
3
1
2
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
S1
0 2 2 2 1 2 2 3 1 2 2 2 1
3
2
5
1
2
2
2
2
5
3
1
3
3
1
2
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
S1
0 2 2 2 1 2 2 3 1 2 2 2 1
3
0 2 2 2 1 2 1 3 0 2 2 2 1
2
5
1
2
2
2
2
5
3
1
3
3
1
2
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
S1
0 2 2 2 1 2 2 3 1 2 2 2 1
3
0 2 2 2 1 2 1 3 0 2 2 2 1
2
5
1
2
2
2
2
5
0 1 2 2 1 2 1 2 0 1 2 2 1
3
1
3
3
1
2
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
S1
0 2 2 2 1 2 2 3 1 2 2 2 1
3
0 2 2 2 1 2 1 3 0 2 2 2 1
2
5
5
2
3
1
3
3
1
2
2
6
1 2 2 1
2
2
0 1 2 2 1 2 1 2 0 1 2 2 1
1 2 2 1 2 1
1
2
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
S1
0 2 2 2 1 2 2 3 1 2 2 2 1
3
0 2 2 2 1 2 1 3 0 2 2 2 1
2
5
5
2
3
1
3
3
1
2
2
6
1 2 2 1
2
2
0 1 2 2 1 2 1 2 0 1 2 2 1
1 2 2 1 2 1
1
2
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
S1
0 2 2 2 1 2 2 3 1 2 2 2 1
3
0 2 2 2 1 2 1 3 0 2 2 2 1
2
5
5
2
3
1
3
3
1
2
2
6
1 2 2 1
2
2
0 1 2 2 1 2 1 2 0 1 2 2 1
1 2 2 1 2 1
1
2
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
3
2
5
1
2
2
2
2
5
3
1
3
3
1
2-1
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
3
2
5
1
2
2
2
2
5
3
1
3
3
1
1
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
3
2
5
1
2
2
2
2
5
3
1
3
3
1
1
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
3
2
5
1
2
2
2
2
5
3
1
3
3
1
1
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
3
2
5
1
2
2
2
2
5
3
1
3
3
1
1
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
3
2
5
1
2
2
2
2
5
3
1
3
3
1
1
2
6
2
2
2
2
2-0 2
2
2
1
1
52
Optimal Target Set
3
2
5
1
2
2
2
2
5
3
1
3
3
1
1
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
3
2
5
1
2
2
2
2
5
3
1
3
3
1
1
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
3
2
5
1
2
2
2
2
5
3
1
3
3
1
1
2
6
2
2
2
2
2
2
2
2
1
1
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Optimal Target Set
3
2
5
1
2
2
2
2
5
3
1
3
1
1
2
2
2
2
2
1
2
2
2
2
2
2
1
2
1
2
6
2
2
5
1
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Toroidal mesh, Torus cordalis, Torus serpentinus
(1,1)
(1,2)
(1,3)
(1,1)
(2,1)
(2,2)
(3,2)
(4,2)
(2,2)
(3,3)
(1,2)
(2,3)
(3,2)
(1,3)
(2,3)
(2,1)
(3,3)
(3,3)
(3,1)
(3,2)
(4,3)
(4,3)
(4,1)
toroidal mesh
(1,1)
(2,2)
(3,1)
(4,1)
(1,3)
(2,3)
(2,1)
(3,1)
(1,2)
(4,2)
torus cordalis
(4,3)
(4,1)
(4,2)
torus serpentinus
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Feedback vertex set (Decycling set)
A subset S of V(G) is a feedback vertex set (or a decycling set ) of a
graph G if the subgraph of G induced by the vertices in V(G) \ S is acyclic.
The size of a minimum feedback vertex set in a graph G is called the
decycling number of G and is denoted by  (G ) .
[Dreyer and Roberts, 2009]
For a vertex subset S of a (k+1)-regular graph G, the target set S can
influence all vertices of V(G) \ S in the social network (G,k) if and only if
S is a feedback vertex set of G.
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[Flocchini et al, 2004]
[Pike and Zou, 2005]
Theorem Let m  3 and n  3 be integers. Then
70
Our Results on Torus cordalis
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Our Results on Torus serpentinus
72
Honeycomb Mesh HMt
HM1
HM 2
HM3
73
Honeycomb Torus HTt
HT1
HT2
HT3
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Honeycomb Rectangular Torus
[Stojmenovic, 1997]
HReT(m, n)
(0,5)
(1,5)
(2,5)
(3,5)
(0,4)
(1,4)
(2,4)
(3,4)
(0,3)
(1,3)
(2,3)
(3,3)
(0,2)
(1,2)
(2,2)
(3,2)
(0,1)
(1,1)
(2,1)
(3,1)
(0,0)
(1,0)
(2,0)
(3,0)
HReT(4,6)
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Honeycomb Rhombic Torus
[Stojmenovic, 1997]
(4,9)
HRoT(m, n)
HRoT(5,6)
(3,8)
(4,8)
(2,7)
(3,7)
(4,7)
(1,6)
(2,6)
(3,6)
(4,6)
(0,5)
(1,5)
(2,5)
(3,5)
(4,5)
(0,4)
(1,4)
(2,4)
(3,4)
(4,4)
(0,3)
(1,3)
(2,3)
(3,3)
(0,2)
(1,2)
(2,2)
(0,1)
(1,1)
78
(0,0)
Generalized Honeycomb (Rectangular) Torus
[Cho and Hsu, 2003]
GHT(m, n, d )
(0,5)
(1,5)
(2,5)
(3,5)
(0,4)
(1,4)
(2,4)
(3,4)
(0,3)
(1,3)
(2,3)
(3,3)
(0,2)
(1,2)
(2,2)
(3,2)
(0,1)
(1,1)
(2,1)
(3,1)
(0,0)
(1,0)
(2,0)
(3,0)
GHT(4, 6, 2)
79
Generalized Honeycomb (Rectangular) Torus
80
Generalized Honeycomb (Rectangular) Torus
V (G )  mn,
E (G) 
3mn
,   3,   1
2
3mn
 (mn  1)
mn  2
2
min-seed(G,   ) 

3 1
4
81
Generalized Honeycomb (Rectangular) Torus
m: odd, n=4t+2
82
Isomorphic to GHT
By definitions,
HReT(m, n) is isomorphic to GHT(m, n, 0).
HRoT(m, n) is isomorphic to GHT(m, n, m (mod n)).
[Cho and Hsu, 2003]
HTt is isomorphic to GHT(t, 6t , 3t ).
84
Hexagonal Grids
planar hexagonal grid
PHG(m,n)
(1,7)
(4,7)
cylindrical hexagonal grid
CHG(m,n)
(0,7)
(4,7)
toroidal hexagonal grid
THG(m,n)
(0,7)
(4,7)
(0,6)
(4,1)
(0,0)
(3,0)
PHG(5,8)
(0,0)
(4,0)
CHG(5,8)
(0,0)
(4,0)
THG(5,8)
85
86
Results on Hexagonal Grids
87
Results on Hexagonal Grids

88