  Many problems in real life can be converted to combinatorial optimization
problems (COPs) on graphs, that is to find a best node state configuration or a
network structure such that the designed objective function is optimized under
some constraints. However, these problems are notorious for their hardness to
solve because most of them are NP-hard or NP-complete. Although traditional
general methods such as simulated annealing (SA), genetic algorithms (GA) and
so forth have been devised to these hard problems, their accuracy and time
consumption are not satisfying in practice. In this work, we proposed a simple,
fast, and general algorithm framework called Gumbel-softmax Optimization (GSO)
for COPs. By introducing Gumbel-softmax technique which is developed in machine
learning community, we can optimize the objective function directly by gradient
descent algorithm regardless of the discrete nature of variables. We test our
algorithm on four different problems including Sherrington-Kirkpatrick (SK)
model, maximum independent set (MIS) problem, modularity optimization, and
structural optimization problem. High-quality solutions can be obtained with
much less time consuming compared to traditional approaches.
