Power management is one of the key research challenges on the path to exascale. Supercomputers today are designed to be worst-case power provisioned, leading to two main problems| limited application performance and under-utilization of procured power. In this paper, we propose RMAP, a practical, low-overhead resource manager targeted at future power-constrained clusters. The goals for RMAP are to improve application performance as well as system power utilization, and thus minimize the average turnaround time for all jobs. Within RMAP, we design and analyze an adaptive policy, which derives job-level power bounds in a fair-share manner and supports overprovisioning and power-aware backfilling. Our results show that our new policy increases system power utilization while adhering to strict job-level power bounds and leads to 31% (19% on average) and 54% (36% on average) faster average turnaround time when compared to worstcase provisioning and naive overprovisioning respectively.