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The problem of influence maximization deals with choosing the optimal set of nodes in a social networks so as to maximize the resulting spread of a technology (opinion, product ownership and so on), given a model of diffusion of influence in a network. A natural extension is a competitive setting, in which the goal is to maximize the spread of our technology in the presence of one or more competitors.

We suggest several natural extensions to the well-studied linear threshold model, showing that the original greedy approach cannot be used.

Furthermore, we show that for a broad family of competitive influence models, it is NP-hard to achieve an approximation that is better than a square root of the optimal solution; the same proof can also be applied to give a negative result for a conjecture in Carnes et al. about a general cascade model for competitive diffusion.

Finally, we suggest a natural model that is amenable to the greedy approach.