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Graph matching involves combinatorial optimization based on edge-to-edgeaffinity matrix, which can be generally formulated as Lawler's QuadraticAssignment Problem (QAP). This paper presents a QAP network directly learningwith the affinity matrix (equivalently the association graph) whereby thematching problem is translated into a constrained vertex classification task.The association graph is learned by an embedding network for vertexclassification, followed by Sinkhorn normalization and a cross-entropy loss forend-to-end learning. We further improve the embedding model on associationgraph by introducing Sinkhorn based matching-aware constraint, as well as dummynodes to deal with unequal sizes of graphs. To our best knowledge, this is oneof the first network to directly learn with the general Lawler's QAP. Incontrast, recent deep matching methods focus on the learning of node/edgefeatures in two graphs respectively. We also show how to extend our network tohypergraph matching, and matching of multiple graphs. Experimental results onboth synthetic graphs and real-world images show its effectiveness. For pureQAP tasks on synthetic data and QAPLIB benchmark, our method can performcompetitively and even surpass state-of-the-art graph matching and QAP solverswith notable less time cost. We provide a project homepage athttp://thinklab.sjtu.edu.cn/project/NGM/index.html.

Neural Graph Matching Network: Learning Lawler's Quadratic Assignment Problem with Extension to Hypergraph and Multiple-graph Matching