Given a directed graph (representing a social network), the influence maximization problem is to find k nodes which, when influenced (or activated), would maximize the number of remaining nodes that get activated. In this paper, we consider a more general version of the problem that includes an additional set of nodes, termed as physical nodes, such that a node in the social network is covered by one or more physical nodes. A physical node exists in one of two states at any time, opened or closed, and there is a constraint on the maximum number of physical nodes that can be opened. In this setting, an inactive node in the social network becomes active if it has enough active neighbors in the social network and if it is covered by at least one of the opened physical nodes. This problem arises in disaster recovery, where a displaced social group decides to return after a disaster only after enough groups in its social network return and some infrastructure components in its neighborhood are repaired. The general problem is NP-hard to approximate within any constant factor and thus we characterize optimal and approximation algorithms for special instances of the problem.
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