  We present new min-max relations in digraphs between the number of paths
satisfying certain conditions and the order of the corresponding cuts. We
define these objects in order to capture, in the context of solving the
half-integral linkage problem, the essential properties needed for reaching a
large bramble of congestion two (or any other constant) from the terminal set.
This strategy has been used ad-hoc in several articles, usually with lengthy
technical proofs, and our objective is to abstract it to make it applicable in
a simpler and unified way. We provide two proofs of the min-max relations, one
consisting in applying Menger's Theorem on appropriately defined auxiliary
digraphs, and an alternative simpler one using matroids, however with worse
polynomial running time.
  As an application, we manage to simplify and improve several results of
Edwards et al. [ESA 2017] and of Giannopoulou et al. [SODA 2022] about finding
half-integral linkages in digraphs. Concerning the former, besides being
simpler, our proof provides an almost optimal bound on the strong connectivity
of a digraph for it to be half-integrally feasible under the presence of a
large bramble of congestion two (or equivalently, if the directed tree-width is
large, which is the hard case). Concerning the latter, our proof uses brambles
as rerouting objects instead of cylindrical grids, hence yielding much better
bounds and being somehow independent of a particular topology.
  We hope that our min-max relations will find further applications as, in our
opinion, they are simple, robust, and versatile to be easily applicable to
different types of routing problems in digraphs.
