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Graphs have been widely used to represent complex data in many applications.Efficient and effective analysis of graphs is important for graph-basedapplications. However, most graph analysis tasks are combinatorial optimization(CO) problems, which are NP-hard. Recent studies have focused a lot on thepotential of using machine learning (ML) to solve graph-based CO problems. Mostrecent methods follow the two-stage framework. The first stage is graphrepresentation learning, which embeds the graphs into low-dimension vectors.The second stage uses ML to solve the CO problems using the embeddings of thegraphs learned in the first stage. The works for the first stage can beclassified into two categories, graph embedding (GE) methods and end-to-end(E2E) learning methods. For GE methods, learning graph embedding has its ownobjective, which may not rely on the CO problems to be solved. The CO problemsare solved by independent downstream tasks. For E2E learning methods, thelearning of graph embeddings does not have its own objective and is anintermediate step of the learning procedure of solving the CO problems. Theworks for the second stage can also be classified into two categories,non-autoregressive methods and autoregressive methods. Non-autoregressivemethods predict a solution for a CO problem in one shot. A non-autoregressivemethod predicts a matrix that denotes the probability of each node/edge being apart of a solution of the CO problem. The solution can be computed from thematrix. Autoregressive methods iteratively extend a partial solution step bystep. At each step, an autoregressive method predicts a node/edge conditionedto current partial solution, which is used to its extension. In this survey, weprovide a thorough overview of recent studies of the graph learning-based COmethods. The survey ends with several remarks on future research directions.

Graph Learning for Combinatorial Optimization: A Survey of State-of-the-Art