In this paper we propose a novel approach to decentralised coordination, that is able to efficiently compute solutions with a guaranteed approximation ratio. Our approach is based on a factor graph representation of the constraint network. It builds a tree structure by eliminating dependencies between the functions and variables within the factor graph that have the least impact on solution quality. It then uses the max-sum algorithm to optimally solve the resulting tree structured constraint network, and provides a bounded approximation specific to the particular problem instance. In addition, we present two generic pruning techniques to reduce the amount of computation that agents must perform when using the max-sum algorithm. When this is combined with the above mentioned approximation algorithm, the agents are able to solve decentralised coordination problems that have very large action spaces with a low computation and communication overhead. We empirically evaluate our approach in a mobile sensor domain, where mobile agents are used to monitor and predict the state of spatial phenomena (e.g., temperature or gas concentration). Such sensors need to coordinate their movements with their direct neighbours to maximise the collective information gain, while predicting measurements at unobserved locations. When applied in this domain, our approach is able to provide solutions which are guaranteed to be within 2% of the optimal solution. Moreover, the two pruning techniques are extremely effective in decreasing the computational effort of each agent by reducing the size of the search space by up to 92%.