  Algorithms for computing or approximating optimal decompositions for
decompositional parameters such as treewidth or clique-width have so far
traditionally been tailored to specific width parameters. Moreover, for
mim-width, no efficient algorithms for computing good decompositions were
known, even under highly restrictive parameterizations. In this work we
identify F-branchwidth as a class of generic decompositional parameters that
can capture mim-width, treewidth, clique-width as well as other measures. We
show that while there is an infinite number of F-branchwidth parameters, only a
handful of these are asymptotically distinct. We then develop fixed-parameter
and kernelization algorithms (under several structural parameterizations) that
can compute every possible F-branchwidth, providing a unifying framework that
can efficiently obtain near-optimal tree-decompositions, k-expressions, as well
as optimal mim-width decompositions.
