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.

翻译：我们提出了有向图中满足特定条件的路径数量与对应割集阶数之间的新极小-极大关系。我们定义这些对象的目标在于，在半整数连通问题的背景下，捕捉从终端集到达拥塞为二（或任意常数）的大荆棘所需的关键性质。该策略已在多篇文章中被临时采用，并伴随冗长的技术性证明，而我们的目标是将其抽象化，使其能以更简单统一的方式应用。我们给出了该极小-极大关系的两种证明：一种通过对适当定义的辅助有向图应用门格尔定理实现；另一种则采用拟阵的替代简化方法，但多项式运行时间稍差。作为应用，我们简化并改进了Edwards等人[ESA 2017]与Giannopoulou等人[SODA 2022]关于有向图中半整数连通问题的部分结果。对于前者，除证明更简单外，我们的方法为有向图在拥塞为二的大荆棘（等价于有向树宽较大时的困难情形）存在时实现半整数可行性提供了近乎最优的强连通性界。对于后者，我们的证明采用荆棘而非圆柱网格作为重路由对象，从而获得显著更优的界并在本质上独立于特定拓扑结构。我们期待这些极小-极大关系能获得更广泛应用，因为其简洁、稳健且灵活，易于适用于有向图中不同类型的路径规划问题。