Approximation and Inapproximation for the Influence Maximization Problem in Social Networks under Deterministic Linear Threshold Model

Abstract

Influence Maximization is the problem of finding a certain amount of people in a social network such that their aggregation influence through the network is maximized. In the past this problem has been widely studied under a number of different models. In 2003, Kempe \emph{et al.} gave a $(1-{1 \over e})$-approximation algorithm for the \emph{linear threshold model} and the \emph{independent cascade model}, which are the two main models in the social network analysis. In addition, Chen \emph{et al.} proved that the problem of exactly computing the influence given a seed set in the two models is $\#$P-hard. Both the \emph{linear threshold model} and the \emph{independent cascade model} are based on randomized propagation. However such information might be obtained by surveys or data mining techniques, which makes great difference on the properties of the problem. In this paper, we study the Influence Maximization problem in the \emph{deterministic linear threshold model}. As a contrast, we show that in the \emph{deterministic linear threshold model}, there is no polynomial time $n^{1-\epsilon}$-approximation unless P=NP even at the simple case that one person needs at most two active neighbors to become active. This inapproximability result is derived with self-contained proofs without using PCP theorem. In the case that a person can be activated when one of its neighbors become active, there is a polynomial time ${e\over e-1}$-approximation, and we prove it is the best possible approximation under a reasonable assumption in the complexity theory, $NP \not\subset DTIME(n^{\log\log n})$. We also show that the exact computation of the final influence given a seed set can be solved in linear time in the \emph{deterministic linear threshold model}. The Least Seed Set problem, which aims to find a seed set with least number of people to activate all the required people in a given social network, is discussed. Using an analysis framework based on Set Cover, we show a $O($log$n)$-approximation in the case that a people become active when one of its neighbors is activated.