Augmented reality (AR) games, particularly those designed for head-mounted displays, have grown increasingly prevalent. However, most existing systems depend on pre-scanned, static environments and rely heavily on continuous tracking or marker-based solutions, which limit adaptability in dynamic physical spaces. This is particularly problematic for AR headsets and glasses, which typically follow the user's head movement and cannot maintain a fixed, stationary view of the scene. Moreover, continuous scene observation is neither power-efficient nor practical for wearable devices, given their limited battery and processing capabilities. A persistent challenge arises when multiple identical objects are present in the environment-standard object tracking pipelines often fail to maintain consistent identities without uninterrupted observation or external sensors. These limitations hinder fluid physical-virtual interactions, especially in dynamic or occluded scenes where continuous tracking is infeasible. To address this, we introduce a novel optimization-based framework for re-identifying identical objects in AR scenes using only one partial egocentric observation frame captured by a headset. We formulate the problem as a label assignment task solved via integer programming, augmented with a Voronoi diagram-based pruning strategy to improve computational efficiency. This method reduces computation time by 50% while preserving 91% accuracy in simulated experiments. Moreover, we evaluated our approach in quantitative synthetic and quantitative real-world experiments. We also conducted three qualitative real-world experiments to demonstrate the practical utility and generalizability for enabling dynamic, markerless object interaction in AR environments. Our video demo is available at https://youtu.be/RwptEfLtW1U.