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Subgraph similarity search, one of the core problems in graph search, concerns whether a target graph approximately contains a query graph. The problem is recently touched by neural methods. However, current neural methods do not consider pruning the target graph, though pruning is critically important in traditional calculations of subgraph similarities. One obstacle to applying pruning in neural methods is {the discrete property of pruning}. In this work, we convert graph pruning to a problem of node relabeling and then relax it to a differentiable problem. Based on this idea, we further design a novel neural network to approximate a type of subgraph distance: the subgraph edit distance (SED). {In particular, we construct the pruning component using a neural structure, and the entire model can be optimized end-to-end.} In the design of the model, we propose an attention mechanism to leverage the information about the query graph and guide the pruning of the target graph. Moreover, we develop a multi-head pruning strategy such that the model can better explore multiple ways of pruning the target graph. The proposed model establishes new state-of-the-art results across seven benchmark datasets. Extensive analysis of the model indicates that the proposed model can reasonably prune the target graph for SED computation. The implementation of our algorithm is released at our Github repo: https://github.com/tufts-ml/Prune4SED.