With the surging popularity of approximate near-neighbor search (ANNS), driven by advances in neural representation learning, the ability to serve queries accompanied by a set of constraints has become an area of intense interest. While the community has recently proposed several algorithms for constrained ANNS, almost all of these methods focus on integration with graph-based indexes, the predominant class of algorithms achieving state-of-the-art performance in latency-recall tradeoffs. In this work, we take a different approach and focus on developing a constrained ANNS algorithm via space partitioning as opposed to graphs. To that end, we introduce Constrained Approximate Partitioned Search (CAPS), an index for ANNS with filters via space partitions that not only retains the benefits of a partition-based algorithm but also outperforms state-of-the-art graph-based constrained search techniques in recall-latency tradeoffs, with only 10% of the index size.