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The growing number of network functions drives the need to install increasing numbers of fine-grained packet classification rules in the network switches. However, this demand for rules is outstripping the size of switch memory. While much work has focused on compressing classification rules, most of this work proposes solutions operating in the memory of a single switch. This paper proposed, instead, a collaborative approach encompassing switches and network functions: we couple approximate classification at switches with fine-grained filtering where needed at network functions to accomplish overall classification. This architecture enables a trade-off between usage of (expensive) switch memory and (cheaper) downstream network bandwidth and network function resources. Our implementation uses approximate classification and Prefiltering to reduce switch memory usage. Our system can reduce memory significantly compared to a strawman approach, as shown by simulations of real traffic traces and rules.