Hardness of computing width parameters based on branch decompositions over the vertex set

Many width parameters of graphs are defined using branch decompositions over the vertex set of the graph and a corresponding cut-function. In this paper, we give a general framework for showing hardness of many width parameters defined in such a way, by reducing from the problem of deciding the exact value of the cut-function. We show that this implies NP-hardness for deciding both booleanwidth and mim-width, and that mim-width is W[1]-hard, and not in APX unless NP=ZPP.