As deep learning models continue to increase in size, the memory requirements for training have surged. While high-level techniques like offloading, recomputation, and compression can alleviate memory pressure, they also introduce overheads. However, a memory-efficient execution plan that includes a reasonable operator execution order and tensor memory layout can significantly increase the models' memory efficiency and reduce overheads from high-level techniques. In this paper, we propose ROAM which operates on computation graph level to derive memory-efficient execution plan with optimized operator order and tensor memory layout for models. We first propose sophisticated theories that carefully consider model structure and training memory load to support optimization for large complex graphs that have not been well supported in the past. An efficient tree-based algorithm is further proposed to search task divisions automatically, along with delivering high performance and effectiveness to solve the problem. Experiments show that ROAM achieves a substantial memory reduction of 35.7%, 13.3%, and 27.2% compared to Pytorch and two state-of-the-art methods and offers a remarkable 53.7x speedup. The evaluation conducted on the expansive GPT2-XL further validates ROAM's scalability.