In urban analytics and smart city analytics, high-resolution Point of Interest (POI) data is indispensable. While commercial APIs like Google Places offer more comprehensive datasets, their high pay-as-you-go costs create a significant barrier for budget-constrained researchers. Existing spatial sampling strategies, such as the fixed grid approach, face an inherent trade-off: they either incur prohibitive costs due to redundant calls in sparse areas or suffer from reduced recall (data omissions) in dense urban cores. To address this, we reframe the data retrieval process as a capacitated facility location problem with unknown demand. Based on this, we propose an adaptive greedy sampling algorithm rooted in the geometric set cover problem, optimized for the constraints of the Google Places API. By using the distance to the farthest POI as the “effective radius,” this algorithm dynamically excludes areas covered by the feedback, thereby geometrically guaranteeing a 100% retrieval rate. Experimental results from Shibuya City (high density) and Chiba City (medium density) in Japan demonstrate that the proposed method can reduce the total number of API calls by approximately 60% compared to the best-performing conventional method (adaptive quadtree), and by up to 99% compared to high-resolution fixed grid baselines. This study aims to contribute to the democratization of high-resolution urban analysis for researchers and practitioners worldwide by minimizing data retrieval costs. The complete code is released as an open-source Python repository on GitHub.