It's known that many problems (e.g. XOR) have the exact solutions represented by neural networks. The question is: What kind of graph theory problems can be solved using neural networks?
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Here is an example that has found a real-world application, in the context of quantum error correction: The decoding of stabilizer codes is a problem of minimum weight-perfect matching on a graph (for the surface code or toric code) or a hypergraph (for the color code). Recurrent neural networks offer a performance that is comparable or better than existing algorithms. Some pointers to the literature: