Epidemic spreading occurs among animals, humans, or computers and causes substantial societal, personal, or economic losses if left undetected. Based on known temporal contact networks, we propose an outbreak detection method that identifies a small set of nodes such that the likelihood of detecting recent outbreaks is maximal. The two-step procedure involves (i) simulating spreading scenarios from all possible seed configurations and (ii) greedily selecting nodes for monitoring in order to maximize the detection likelihood. We find that the detection likelihood is a submodular set function for which it has been proven that greedy optimization attains at least 63% of the optimal (intractable) solution. The results show that the proposed method detects more outbreaks than benchmark methods suggested recently and is robust against badly chosen parameters. In addition, our method can be used for outbreak source detection. A limitation of this method is its heavy use of computational resources. However, for large graphs the method could be easily parallelized.