Paper Introduction: Combinatorial Model and Bounds for Target Set Selection

Preliminaries
A Combinatorial Model for TSS
Combinatorial Bounds for Perfect TSS
Paper Introduction:
Combinatorial Model and Bounds for Target Set
Selection
Yu Liu
NII Weekly Seminar
April 21, 2014
Yu Liu Paper Introduction: Combinatorial Model and Bounds for Target

Preliminaries
About The Paper
Eyal Ackerman, Oren Ben-Zwi, Guy Wolfovitz: Combinatorial
model and bounds for target set selection. Theor. Comput.
Sci. 411(44-46): 4017-4022 (2010).
a combinatorial model for the dynamic activation process of
inﬂuential networks;
representing Perfect Target Set Selection Problem and its
variants by linear integer programs;
combinatorial lower and upper bounds on the size of the
minimum Perfect Target Set

Preliminaries
inﬂuential networks
diﬀusion models/process

Preliminaries
Influential Network
An influential network represents how the adoption of individuals’
decisions are influenced by recommendations from their friends.
1
2
3
4
5

Preliminaries
Diﬀusion Models/Process under Threshold Model
Let G = (V , E) be a digraph, S ⊆ V , t : V → N be a threshold
function.
Activation process
is a chain of vertex subsets:
Active[0] ⊂ Active[1] ⊂ Active[2] ⊂ . . . Active[z] ⊆ V .
Where Active[0] = S, ∀i > 0, Active[i] = Active[i − 1] ∪ {u|t(u) ≤
|{v ∈ Active[i − 1]|(v, u) ∈ E}|}
Active[S] =: Active[z], means S can activate set Active[z]

Preliminaries
Diﬀusion Models/Process under Threshold Model
suppose θ1 = 2, θ2 = 1, θ3 = 1, θ4 = 1, θ5 = 3
1
2
3
4
5
Active[0] = {1}, Active[1] = {1, 2, 3}, Active[2] =
{1, 2, 3, 4}, Active[3] = {1, 2, 3, 4, 5}
this paper only discuss monotone activation process

Preliminaries
Target Set Selection Problem
input: two integers k, l and a digraph G = (V , E) with
thresholds t : V → N.
output: a set S ⊆ V , such that |S| ≤ k, |Active[S]| ≥ l.
When l = |V | and t(v) = deg(v) it reduces to Vertex Cover
problem.

Preliminaries
Minimum Target Set
input: an integers l and a digraph G = (V , E) with thresholds
t : V → N.
output: a smallest set S ⊆ V , such that |Active[S]| ≥ l.

Preliminaries
Maximum Active Set
input: an integers k and a digraph G = (V , E) with
output: a set S ⊆ V and |S| = k , such that ∀S ⊆ V and
S = S, |Active[S]| ≥ |Active[S ]|.

Preliminaries
Inapproximability of TSS
Target Set Selection problem is NP-hard, and hard to approximate.
NP-hard to approximate within a factor of n1− , for any
constant > 0 [KKT03].
For the special case where the thresholds are taken uniformly
at random, a constant-factor approximation algorithm
exists[MR07].
No O(2log1− |V |)-approximation algorithm exists, for any
constant > 0, even when all thresholds are set to 2 and the
graph is of constant degree [Che08].

Preliminaries
Some Terminologies
A target set is called perfect if it activates the entire graph.
majority threshold : ∀v ∈ V , t(v) ≥ deg(v)/2
strict majority threshold : ∀v ∈ V , t(v) > deg(v)/2
irreversible dynamic monopoly (dynamo) usually refers to a
perfect target set under majority or strict majority thresholds
G[A] means a subgraph of G induced by a vertex set A

Preliminaries
Main Results
1 bounds on the size of the minimum perfect target set:
the size of the minimum perfect target set is at most 2|V |/3
under strict majority thresholds, for both directed and
undirected graphs
an upper bound of |V |/2 on the size of the minimum perfect
target set under majority thresholds, both for directed and
undirected graphs.
2 a trivial constant factor approximation exists if
∆(G)/δ(G) is bounded (∆(G) and δ(G) are the maximum
and minimum degrees in G,respectively)

Preliminaries
Combinatorial Target Set Selection
Combinatorial Target Set Selection
input: two integers k, l and a digraph G = (V , E) with
output: a set S ⊆ V , such that |S| ≤ k and there is a set
A ⊆ V such that S ⊆ A, |A| ≥ l, and one can remove edges
such that G[A] is acyclic and degin(v) ≥ t(v) for every vertex
v ∈ A S.
S is a solution of Target Set Selection iﬀ it is a solution of
Combinatorial Target Set Selection.
under this model, we can represent the optimization problems
as integer linear programs (IP)

Preliminaries
IP for Minimum Target Set
The number of variables is Θ(n2) and the number of constraints is
Θ(n3).

Preliminaries
IP for Maximum Target Set
The number of variables is Θ(n2) and the number of constraints is
Θ(n3).

Preliminaries
Combinatorial Bounds for Strict Majority Thresholds
For a digraph G(V , E), ∀v ∈ V degin(v) > 0,
there is an algorithm which ﬁnds in polynomial time a target set of
size at most 2n/3

Preliminaries
Combinatorial Bounds for Majority Thresholds
For a digraph G(V , E), ∀v ∈ V degin(v) > 0,
there is an algorithm which ﬁnds in polynomial time a target set of
size at most n/2

Preliminaries
Proof
Suppose A is activated by some S, then S can be found as follows:
π is a random permutation of the vertices in A;
remove edges in G[A] that violate the order of vertices, i.e.,
(u, v), such that π(u) > π(v);
then ∀v ∈ A, v ∈ S or degin(v) ≥ t(v).

Preliminaries
Proof
The expected number of vertices in S is:
E[|S|] =
v∈A
t(v)
degin(v) + 1
(1)
(1) gives an upper bound on the size of S.
for the Minimum Perfect Target Set Problem, A = V .

Preliminaries
Proof
E[|S|] =
v∈A
t(v)
degin(v) + 1
(1)
when t(v) = degin(v)+1
2 , t(v)
degin(v)+1 ≥ 2/3,
when t(v) = degin(v)
2 , t(v)
degin(v)+1 ≥ 1/2

Preliminaries
More-than-majority thresholds
For undirected graph, if ∀v ∈ V , t(v) ≥ deg(v)/2 + 1, then
the minimum perfect target set is at least |V |/T, where
T = maxv∈V t(v).
ﬁnds a perfect target set of size at most 2|V |/3 is a
2T/3-approximation.

Preliminaries
Proof
Let undirected G=(V,E), z be the smallest integer such that
Active[z] = V . For every i, 0 ≤ i ≤ z, deﬁne a potential function
Φ(i) =
v /∈Active[i]
t(v) + |E[Active[i]]|,
where E[U] denotes the set of edges in E induced by set U.

Preliminaries
Proof
Finally we have:
|E| + |V | v∈V
deg(v)
2 + 1
v∈V t(v)
v /∈S t(v) + v∈S t(v)
|E| + v∈S t(v)
Thus, |V | ≤ v∈S t(v) .

Preliminaries
Extention for More General Settings
When t(v) = θ ∗ deg(v) , for a constant θ ∈ (1/2, 1],
a lower bound on |S|:
|V |δ(G)(2θ − 1)
∆(G) + δ(G)(2θ − 1)
an approximation ratio:
θ(∆(G) + δ(G)(2θ − 1))
δ(G)(2θ − 1)
= O(
∆(G)
δ(G)
)

Preliminaries
Summary
Simper proof of inapproximability results that same as
[Che08], for TSS
Integer linear programs for TSS
Tighter upper bound for majority / strict majority thresholds
2|V |/3 compared with 0.727|V | (digraphs) and
0.7732|V |(undirected graphs) in [CL13], for strict majority
thresholds

Preliminaries
Bibliography
Ning Chen.
On the approximability of influence in social networks.
In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’08, pages
1029–1037, Philadelphia, PA, USA, 2008. Society for Industrial and Applied Mathematics.
Ching-Lueh Chang and Yuh-Dauh Lyuu.
Bounding the sizes of dynamic monopolies and convergent sets for threshold-based cascades.
Theor. Comput. Sci., 468:37–49, January 2013.
David Kempe, Jon Kleinberg, and Éva Tardos.
Maximizing the spread of influence through a social network.
In Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data
Mining, KDD ’03, pages 137–146, New York, NY, USA, 2003. ACM.
Elchanan Mossel and Sebastien Roch.
On the submodularity of influence in social networks.
In Proceedings of the Thirty-ninth Annual ACM Symposium on Theory of Computing, STOC ’07, pages
128–134, New York, NY, USA, 2007. ACM.

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