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We present A-GAP, a novel protocol for continuous monitoring of network state variables, which aims at achieving a given monitoring accuracy with minimal overhead. Network state variables are computed from device counters using aggregation functions, such as SUM, AVERAGE and MAX. The accuracy objective is expressed as the average estimation error. A-GAP is decentralized and asynchronous to achieve robustness and scalability. It executes on an overlay that interconnects management processes on the devices. On this overlay, the protocol maintains a spanning tree and updates the network state variables through incremental aggregation. It dynamically configures local filters that control whether an update is sent towards the root of the tree. It reduces the overhead by attempting to minimize the maximum processing load over all management processes. We evaluate A-GAP through simulation using an ISP topology and real traces. The results show that we can effectively control the trade-off between accuracy and protocol overhead, that the overhead can be reduced significantly by allowing small errors, and that an accurate estimation of the error distribution can be provided in real-time.