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