  The minimum cost multicut problem is the NP-hard/APX-hard combinatorial
optimization problem of partitioning a real-valued edge-weighted graph such as
to minimize the total cost of the partition. While graph convolutional neural
networks (GNN) have proven to be promising in the context of combinatorial
optimization, most of them are only tailored to or tested on positive-valued
edge weights, i.e. they do not comply to the nature of the multicut problem. We
therefore adapt various GNN architectures including Graph Convolutional
Networks, Signed Graph Convolutional Networks and Graph Isomorphic Networks to
facilitate the efficient encoding of real-valued edge costs. Moreover, we
employ a reformulation of the multicut ILP constraints to a polynomial program
as loss function that allows to learn feasible multicut solutions in a scalable
way. Thus, we provide the first approach towards end-to-end trainable
multicuts. Our findings support that GNN approaches can produce good solutions
in practice while providing lower computation times and largely improved
scalability compared to LP solvers and optimized heuristics, especially when
considering large instances.
