2, and otherwise can be solved via b-matching techniques. When jobs are arbitrary in length and can be preempted, but have a single feasible interval, we also show that a large class of algorithms is 5-approximate. We also empirically compare algorithms within this class with the incumbent greedy algorithm due to Wolsey; the latter algorithm is widely implemented in practice. This comparison is cast within a framework general enough to capture other canonical covering problems, most notably Capacitated Set Cover. In particular, we design a heuristic LPO that essentially complements Wolsey's algorithm and propose optimizations to both approaches. At a high level, our findings solidly establish LPO as a competitive, if not superior, alternative to Wolsey's algorithm, with respect to both solution quality and number of subroutine calls made. Finally, this thesis makes theoretical headway on the well-studied busy time problem. The key assumption that sets this apart from the aforementioned active time problem is that under the busy time model, the system has access to an unlimited number of identical batch machines. For non-preemptive jobs of arbitrary length, we give a 3-approximation that leverages important insights for the special case where the instance consists of interval jobs. When preemption is permitted, we give an exact algorithm for unbounded B; this result yields a simple 2-approximation for bounded B.">