We consider a class of queueing networks referred to as "generalized constrained queueing networks" which form the basis of several different communication networks and information systems. These networks consist of a collection of queues such that only certain sets of queues can be concurrently served. Whenever a queue is served, the system receives a certain reward. Different rewards are obtained for serving different queues, and furthermore, the reward obtained for serving a queue depends on the set of concurrently served queues. We demonstrate that the dependence of the rewards on the schedules alter fundamental relations between performance metrics like throughput and stability. Specifically, maximizing the throughput is no longer equivalent to maximizing the stability region; we therefore need to maximize one subject to certain constraints on the other. Since stability is critical for bounding packet delays and buffer overflow, we focus on maximizing the throughput subject to stabilizing the system. We design provably optimal scheduling strategies that attain this goal by scheduling the queues for service based on the queue lengths and the rewards provided by different selections. The proposed scheduling strategies are however computationally complex. We subsequently develop techniques to reduce the complexity and yet attain the same throughput and stability region. We demonstrate that our framework is general enough to accommodate random rewards and random scheduling constraints.