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