We focus on the election manipulation problem through social influence, where a manipulator exploits a social network to make her most preferred candidate win an election. Influence is due to information in favor of and/or against one or multiple candidates, sent by seeds and spreading through the network according to the independent cascade model. We provide a comprehensive study of the election control problem, investigating two forms of manipulations: seeding to buy influencers given a social network, and removing or adding edges in the social network given the seeds and the information sent. In particular, we study a wide range of cases distinguishing for the number of candidates or the kind of information spread over the network. Our main result is positive for democracy, and it shows that the election manipulation problem is not affordable in the worst-case except for trivial classes of instances, even when one accepts to approximate the margin of victory. In the case of seeding, we also show that the manipulation is hard even if the graph is a line and that a large class of algorithms, including most of the approaches recently adopted for social-influence problems, fail to compute a bounded approximation even on elementary networks, as undirected graphs with every node having a degree at most two or directed trees. In the case of edge removal or addition, our hardness results also apply to the basic case of social influence maximization/minimization. In contrast, the hardness of election manipulation holds even when the manipulator has an unlimited budget, being allowed to remove or add an arbitrary number of edges.
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