In recent years, cloud computing concepts have been extended towards the network edge, leading to paradigms like edge computing and fog computing. As a result, applications can be placed on a variety of resources, including fog nodes and cloud data centers. Application placement has significant impact on important metrics like latency. Finding an optimal application placement is computationally challenging, particularly because of the potentially huge number of infrastructure nodes and application components. To overcome the limited scalability of application placement algorithms, optimization can be decentralized, i.e., performed separately for different parts of the infrastructure. The infrastructure can be split into fog colonies, where a fog colony consists of the computational resources in a given geographical region. Application placement can then be performed for the individual fog colonies, thus mitigating the scalability problem. However, independent optimization of application placement in different fog colonies may lead to missed synergies and thus to sub-optimal overall results. Hence, some kind of coordination between fog colonies may be beneficial. In this paper, we analyze the effects of decentralization and coordination on the optimization results. In particular, we compare empirically four different approaches: (i) centralized decision-making, where decisions are made in one go for the entire infrastructure, (ii) independent fog colonies, where optimization is carried out in each fog colony independently from each other, (iii) fog colonies with communication, where excess application components in one fog colony can be sent to a neighboring fog colony, and (iv) fog colonies with overlaps, where shared resources may be dynamically distributed between neighboring fog colonies. Our experiments show that, for large problem instances, decentralization combined with coordination leads to the best results.