paper introduction: combinatorial model and bounds for target set selection

PreliminariesA 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 Set Selection



About The Paper

Eyal Ackerman, Oren Ben-Zwi, Guy Wolfovitz: Combinatorialmodel and bounds for target set selection. Theor. Comput.Sci. 411(44-46): 4017-4022 (2010).

a combinatorial model for the dynamic activation process ofinfluential networks;

representing Perfect Target Set Selection Problem and itsvariants by linear integer programs;

combinatorial lower and upper bounds on the size of theminimum Perfect Target Set




influential networks

diffusion models/process




Influential Network

An influential network represents how the adoption of individuals’decisions are influenced by recommendations from their friends.

1

2

3

4

5




Diffusion Models/Process under Threshold Model

Let G = (V ,E ) be a digraph, S ⊆ V , t : V → N be a thresholdfunction.

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 ]




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




Target Set Selection Problem

input: two integers k, l and a digraph G = (V ,E ) withthresholds 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 Coverproblem.




Minimum Target Set

input: an integers l and a digraph G = (V ,E ) with thresholdst : V → N.

output: a smallest set S ⊆ V , such that |Active[S ]| ≥ l .




Maximum Active Set

input: an integers k and a digraph G = (V ,E ) withthresholds t : V → N.

output: a set S ⊆ V and |S | = k , such that ∀S ′ ⊆ V andS ′ 6= S , |Active[S ]| ≥ |Active[S ′]|.




Inapproximability of TSS

Target Set Selection problem is NP-hard, and hard to approximate.

NP-hard to approximate within a factor of n1−ε, for anyconstant ε > 0 [KKT03].

For the special case where the thresholds are taken uniformlyat random, a constant-factor approximation algorithmexists[MR07].

No O(2log1−ε|V |)-approximation algorithm exists, for any

constant ε > 0, even when all thresholds are set to 2 and thegraph is of constant degree [Che08].




Some Terminologies

A target set is called perfect if it activates the entire graph.

majority threshold : ∀v ∈ V , t(v) ≥ ddeg(v)/2estrict majority threshold : ∀v ∈ V , t(v) > ddeg(v)/2eirreversible dynamic monopoly (dynamo) usually refers to aperfect target set under majority or strict majority thresholds

G [A] means a subgraph of G induced by a vertex set A




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 |/3under strict majority thresholds, for both directed andundirected graphsan upper bound of |V |/2 on the size of the minimum perfecttarget set under majority thresholds, both for directed andundirected graphs.

2 a trivial constant factor approximation exists if

∆(G )/δ(G ) is bounded (∆(G ) and δ(G ) are the maximumand minimum degrees in G ,respectively)




Combinatorial Target Set Selection

Combinatorial Target Set Selection

input: two integers k, l and a digraph G = (V ,E ) withthresholds t : V → N.

output: a set S ⊆ V , such that |S | ≤ k and there is a setA ⊆ V such that S ⊆ A, |A| ≥ l , and one can remove edgessuch that G [A] is acyclic and degin(v) ≥ t(v) for every vertexv ∈ A \ S .

S is a solution of Target Set Selection iff it is a solution ofCombinatorial Target Set Selection.

under this model, we can represent the optimization problemsas integer linear programs (IP)




IP for Minimum Target Set

The number of variables is Θ(n2) and the number of constraints isΘ(n3).




IP for Maximum Target Set

The number of variables is Θ(n2) and the number of constraints isΘ(n3).




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 ofsize at most 2n/3




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 ofsize at most n/2




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).




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 .




Proof

E[|S |] =∑v∈A

t(v)

degin(v) + 1(1)

when t(v) = ddegin(v)+12 e, t(v)

degin(v)+1 ≥ 2/3,

when t(v) = ddegin(v)2 e, t(v)

degin(v)+1 ≥ 1/2




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 , whereT = maxv∈V t(v).

finds a perfect target set of size at most 2|V |/3 is a2T/3-approximation.




Proof

Let undirected G=(V,E), z be the smallest integer such thatActive[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.




Proof

Finally we have:

|E |+ |V | 6∑

v∈Vdeg(v)

2 + 16

∑v∈V t(v)

6∑

v /∈S t(v) +∑

v∈S t(v)6 |E |+

∑v∈S t(v)

Thus, |V | ≤∑

v∈S t(v) .




Extention for More General Settings

When t(v) = dθ ∗ deg(v)e, 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 ))




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) and0.7732|V |(undirected graphs) in [CL13], for strict majoritythresholds




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, pages1029–1037, 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 Eva Tardos.

Maximizing the spread of influence through a social network.In Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and DataMining, 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, pages128–134, New York, NY, USA, 2007. ACM.


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