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.

The Minimum Cost Multicut Problem (MP) is a popular way for obtaining a graph decomposition by optimizing binary edge labels over edge costs. While the formulation of a MP from independently estimated costs per edge is highly flexible and intuitive, solving the MP is NP-hard and time-expensive. As a remedy, recent work proposed to predict edge probabilities with awareness to potential conflicts by incorporating cycle constraints in the prediction process. We argue that such formulation, while providing a first step towards end-to-end learnable edge weights, is suboptimal, since it is built upon a loose relaxation of the MP. We therefore propose an adaptive CRF that allows to progressively consider more violated constraints and, in consequence, to issue solutions with higher validity. Experiments on the BSDS500 benchmark for natural image segmentation as well as on electron microscopic recordings show that our approach yields more precise edge detection and image segmentation.