Segmenting unseen object instances in unstructured environments is an important skill for robots to perform grasping-related tasks, where the trade-off between efficiency and accuracy is an urgent challenge to be solved. In this work, we propose a fast unseen object instance segmentation (Fast UOIS) method that utilizes predicted center offsets of objects to compute the positions of local maxima and minima, which are then used for selecting initial seed points required by the mean-shift clustering algorithm. This clustering algorithm that adaptively generates seed points can quickly and accurately obtain instance masks of unseen objects. Accordingly, Fast UOIS first generates pixel-wise predictions of object classes and center offsets from synthetic depth images. Then, these predictions are used by the clustering algorithm to calculate initial seed points and to find possible object instances. Finally, the depth information corresponding to the filtered instance masks is fed into the grasp generation network to generate grasp poses. Benchmark experiments show that our method can be well transferred to the real world and can quickly generate sharp and accurate instance masks. Furthermore, we demonstrate that our method is capable of segmenting instance masks of unseen objects for robotic grasping.