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In-memory computing (IMC) with non-volatile memories (NVMs) has emerged as a promising approach to address the rapidly growing computational demands of Deep Neural Networks (DNNs). Mapping DNN layers spatially onto NVM-based IMC accelerators achieves high degrees of parallelism. However, two challenges that arise in this approach are the highly non-uniform distribution of layer processing times and high area requirements. We propose LRMP, a method to jointly apply layer replication and mixed precision quantization to improve the performance of DNNs when mapped to area-constrained NVM-based IMC accelerators. LRMP uses a combination of reinforcement learning and integer linear programming to search the replication-quantization design space using a model that is closely informed by the target hardware architecture. Across five DNN benchmarks, LRMP achieves 2.8-9$\times$ latency and 11.8-19$\times$ throughput improvement at iso-accuracy.