  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.
