Stream grids are wide-area grid computing environments that are fed by a set of stream data sources. Such grids are becoming more wide-spread due to the large scale deployment of sensor networks for a wide range of applications, from monitoring geophysical activities to supply chain management coupled with applications like network monitoring. Queries external to the system arrive on any node in the grid seeking data from one or more data streams. The kind of queries considered in this work are (1) lifetime queries and (2) long running queries where new query arrivals and query revocations are infrequent. From the system perspective, computing the optimal query plan for the set of queries incident on the grid would ensure minimal system-wide resource usage, thereby maximizing the number of concurrent queries that can be supported. The key challenge in such a system is multi-query optimization. In this work, we analyze the complexity of multi-query optimization for select, project and ...