Abstract: Non-maximum suppression (NMS) is an indispensable post-processing step in object detection. With the continuous optimization of network models, NMS has become the last mile'' to enhance the efficiency of object detection. This paper systematically analyzes NMS from a graph theory perspective for the first time, revealing its intrinsic structure. Consequently, we propose two optimization methods, namely QSI-NMS and BOE-NMS. The former is a fast recursive divide-and-conquer algorithm with negligible mAP loss, and its extended version (eQSI-NMS) achieves optimal complexity of $\mathcal{O}(n\log n)$. The latter, concentrating on the locality of NMS, achieves an optimization at a constant level without an mAP loss penalty. Moreover, to facilitate rapid evaluation of NMS methods for researchers, we introduce NMS-Bench, the first benchmark designed to comprehensively assess various NMS methods. Taking the YOLOv8-N model on MS COCO 2017 as the benchmark setup, our method QSI-NMS provides $6.2\times$ speed of original NMS on the benchmark, with a $0.1\%$ decrease in mAP. The optimal eQSI-NMS, with only a $0.3\%$ mAP decrease, achieves $10.7\times$ speed. Meanwhile, BOE-NMS exhibits $5.1\times$ speed with no compromise in mAP.