  In the last few years, graph convolutional networks (GCN) have become a
popular research direction in the machine learning community to tackle NP-hard
combinatorial optimization problems (COPs) defined on graphs. While the
obtained results are usually still not competitive with problem-specific
solution approaches from the operations research community, GCNs often lead to
improvements compared to previous machine learning approaches for classical
COPs such as the traveling salesperson problem (TSP).
  In this work we present a preliminary study on using GCNs for solving the
vertex p-center problem (PCP), which is another classic COP on graphs. In
particular, we investigate whether a successful model based on end-to-end
training for the TSP can be adapted to a PCP, which is defined on a similar 2D
Euclidean graph input as the usually used version of the TSP. However, the
objective of the PCP has a min-max structure which could lead to many symmetric
optimal, i.e., ground-truth solutions and other potential difficulties for
learning. Our obtained preliminary results show that indeed a direct transfer
of network architecture ideas does not seem to work too well. Thus we think
that the PCP could be an interesting benchmark problem for new ideas and
developments in the area of GCNs.
