An Inapproximability Result for the Target Set Selection Problem on Bipartite Graphs

12/04/2018 ∙ by Suman Banerjee, et al. ∙ IIT Kharagpur 0

Given an undirected graph G(V, E, τ) modeling a 'social network', where each node v is associated with a threshold value τ(v), a set of vertices S⊆ V(G) (called 'seed nodes') is chosen initially. Now a 'social contagion process' is defined on G as follows. At time t=0 the nodes in S have some information, and it is diffused in discrete time steps. An 'un-influenced' node v will change its state to 'influenced' at time step t if it has at least τ(v) number of neighbors, which are influenced at time step t-1. The process continues until no more node activation is possible. Based on this diffusion process, a well-studied problem in the literature is the 'Target Set Selection Problem (TSS Problem)', where the goal is to influence all the nodes of the network by initially choosing the minimum number of seed nodes. Chen et al. [On the Approximability of Influence in Social Networks. SIAM Journal on Discrete Mathematics, 23(3):1400-1415, 2009] showed that the decision version of this optimization problem is NP-Hard on bounded degree bipartite graphs. In this paper, we show that this problem on bipartite graph does not admit an approximation algorithm with a performance guarantee asymptotically better than O( n_min), where n_min is the cardinality of the smaller bipartition, unless P=NP.

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1 Introduction

1.1 The Influence Diffusion Process

Diffusion is a natural phenomenon in many real-world networks such as diffusion of information, innovation, ideas, rumors in an Online Social Network [1] [5]; propagation of virus, wormhole in a computer network [8]; spreading of contaminated diseases in a human contact network [11] and many more. Depending on the situation, we want to maximize/minimize the spread. For example, in the case of propagation of information in a social network, sometimes we want to maximize the spread so that a large number of people are aware of the fact. On the other hand in the case of spreading of contaminated disease, we want to minimize the spread. In this paper, the practical essence of our study is in and around the first situation.

In reality, diffusion starts from a set of initial nodes known as seed nodes. A node can be in any one of the following two states: influenced (also known as active) or not influenced (also known as inactive). A node can change its state from inactive to active, however, not the vice versa. Only the seed nodes are active initially and the information is disseminated in discrete time steps from these seed nodes. A node will be influenced at time step , if there are at least number of nodes in its neighborhood, which have been activated at time . The diffusion process stops when no more node-activation is possible.

1.2 Problem Definition

In our study, we assume that the social network is represented as an undirected graph 111Now onwards, we use the term graph and network interchangaebily. , where and are the set of vertices and edges of , respectively. is a threshold function assigning each node to its threshold value, i.e., . Let be a set of seed nodes from where diffusion starts. As described in Section 1.1, influence propagates in discrete time steps, i.e., , where denotes the set of nodes that has been influenced on or before the time stamp and , . For all , the diffusion process can be expressed by the following equation:

The Target Set Selection Problem is a problem based on this diffusion phenomenon, where the goal is to select a minimum cardinality seed set that makes all the nodes of the network influenced. Now, we formally state the optimization version of this problem below. [frametitle=TSS Problem (Optimization Version), style=MyFrame]

Instance: An undirected graph with .

Problem: Find a minimum cardinality seed set such that .

1.3 Related Work

The TSS Problem is a variant of the Social Influence Maximization Problem (SIM Problem) introduced by Kempe et al. [9, 10], where given a positive integer the goal is to choose a -sized seed set that maximizes the number of nodes influenced. Under two probabilistic diffusion models, they show that the SIM Problem is NP-Hard and also hard to approximate within a factor of , for any . The first interesting result on the TSS Problem was initially put forward by Chen et al. [2]. They showed that the TSS Problem cannot be approximated within a factor of of the optimum for a fixed constant , unless , by a reduction from the MINREP Problem. They also show that the TSS Problem is NP-Hard for bounded degree bipartite graphs with a threshold value not greater than at each vertex by a reduction from a variant of the 3-SAT Problem. For trees, they propose a polynomial time algorithm. In [3], Chiang et al. show that the TSS Problem can be solved in linear time for block-cactus graphs with an arbitrary threshold and chordal graph with threshold at most . In [4], Chiang et al. study the TSS Problem on Honeycomb Networks under the majority threshold, where the threshold value of each node is more than half of its degree. They give the exact value for different types of honeycomb networks under strict majority threshold model. Chopin et al. [6] show that upper bounding the threshold to a constant leads to efficiently solvable instances of TSS Problem under the parameterized complexity theoretic framework. They show that the TSS Problem is hard with respect to the combined parameters feedback vertex cover, distance to co-graph, distance to interval graph, pathwidth, cluster vertex deletion number and hard with respect to the parameter seed set cardinality and fixed parameter tractable with respect to the parameters distance to clique and bandwidth. Dvovrak et al. [7] added few more results in the parameterized setting. They showed that TSS Problem is hard with respect to parameters neighborhood diversity and under majority threshold this problem has an FPT algorithm with respect to the parameters neighborhood diversity, twin cover, modular width.

1.4 Our Contribution

We prove the following inapproximability result for the TSS Problem on bipartite graphs.

Theorem 1.

Unless , if the underlying influence graph is bipartite, the TSS Problem cannot have a polynomial time approximation algorithm with a performance guarantee better than a factor of , where is the cardinality of the smaller part in the bipartition.

To the best of the authors’ knowledge, other than Chen et al.’s [2] inapproximability bound of for a fixed constant for the TSS Problem on general graphs with majority threshold, no other inapproximability result is known for any special graph classes. The result presented in this paper is the second one in this direction.

1.5 Symbols and Notations

Throughout the paper, we consider finite, undirected and simple bipartite graphs. We use to denote a bipartite graph, where and denote the bipartition of the vertex set. For any , denotes the cardinality of . For any , denotes the number of edges incident on and denotes the set of adjacent vertices of in . For any positive integer , denotes the set . In this paper, all logarithms considered are to the base , unless mentioned otherwise.

2 Inapproximability Result for the TSS Problem on Bipartite Graphs

We state and prove an auxiliary lemma in Section 2.1 which is required to establish the inapproxibility result of Section 2.2.

2.1 An Auxiliary Lemma

Lemma 1.

Let be a bipartite influence graph with , if and , if . Further, it is given that the degree of every vertex in is at least . Then given any feasible solution for the TSS Problem, we can generate another feasible solution in polynomial time with that satisfies the following two properties: (i) and (ii) , .

Proof.

If the given feasible solution is a subset of , then . Otherwise, we construct from by following the iterative procedure given below. Initialize . We pick a vertex from . If , then pick a vertex from (say, ) and update as . If we perform this operation for all the nodes of , we get a set with . It is easy to observe that this operation can be performed in polynomial time. Since for each node , we choose at most one vertex from to include it in the set , we have, and . Now, we argue that is also a feasible solution.

Using the fact that is a feasible solution for the TSS Problem on , we have the following two observations:

  • Observation 1: , either or .

  • Observation 2: , .

From Observation 2, we can say that , . Also , at least one of its neighbors is in . Hence, we can say . As, . The nodes in will be able to influence all the nodes of . The nodes in in turn influence the nodes in . Hence, is a feasible solution for the TSS Problem on . This completes the proof. ∎

2.2 An Approximation Preserving Reduction from the Set Cover Problem

In this section, we study the TSS Problem on bipartite graphs and obtain an factor inapproximability result by a reduction from the classical set cover problem, where is the cardinality of the smaller part in the bipartition. First, we state the optimization version of the set cover problem. [frametitle=Set Cover Problem (Optimization Version), style=MyFrame]

Instance: A ground set of n elements, a collection of m subsets of .

Question: Find a minimum sized sub-collection such that ?

An incremental greedy approach, which starts with an empty set and in each iteration picks a subset that covers the maximum number of uncovered elements, yields an factor approximation guarantee and this bound is tight.

Theorem 2.

[12] Unless , the set cover problem cannot have a polynomial time approximation algorithm with a performance guarantee better than .

Next, we state our reduction from the set cover problem to the TSS Problem.

Construction 1.

Let be an instance of the set cover problem, where and is a family of subsets of . From this instance of the set cover problem, we construct a bipartite influence graph where and , and , .

Statement of Theorem 1: “Unless , if the underlying influence graph is bipartite, the TSS Problem cannot have a polynomial time approximation algorithm with a performance guarantee better than , where is the cardinality of the smaller part in the bipartition.”

Proof.

We prove this statement by an approximation preserving reduction from the set cover problem. Given an instance of the set cover problem with and , we generate a bipartite influence graph as stated in Construction 1. Let us assume that is a polynomial time algorithm for the TSS Problem having an approximation guarantee better than a factor of , where is the cardinality of the smaller part in the bipartition. We run the algorithm on the bipartite influence graph and obtain a seed set . Now, we construct a seed set from with that satisfies the two properties given in Lemma 1. We know from Lemma 1 that such an can be constructed in polynomial time. Let, . Then, since satisfies Property (ii) of Lemma 1 it is easy to observe that is a set cover of of size at most of a minimum set cover. Hence, Algorithm combined with Construction 1 can be used to solve the set cover problem with an approximation guarantee better than a factor of , which according to Theorem 2 is not possible unless . ∎

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