Given a set of source-sink pairs, the maximum multiflow problem asks for the maximum total amount of flow that can be feasibly routed between them. The minimum multicut, a dual problem to multiflow, seeks the minimum-cost set of edges whose removal disconnects all the source-sink pairs. It is easy to see that the value of the minimum multicut is at least that of the maximum multiflow, and their ratio is called the multiflow-multicut gap. The classical max-flow min-cut theorem states that when there is only one source-sink pair, the gap is exactly one. However, in general, it is well known that this gap can be arbitrarily large. In this paper, we study this gap for classes of planar graphs and establish improved lower bound results. In particular, we show that this gap is at least $\frac{16}{7}$ for the class of planar graphs, improving upon the decades-old lower bound of 2. More importantly, we develop new techniques for proving such a lower bound, which may be useful in other settings as well.

翻译：给定一组源-汇对，最大多流问题旨在求解这些节点间可路由的最大总流量。最小多割作为多流问题的对偶问题，则寻求断开所有源-汇连接所需的最小代价边集。显然最小多割的值至少等于最大多流的值，二者的比值称为多流-多割间隙。经典的最大流最小割定理表明当仅存在单一源-汇对时，该间隙恰好为1。然而在一般情况下，该间隙可能任意大已是公认结论。本文针对平面图类研究该间隙，建立了改进的下界结果。特别地，我们证明对于平面图类该间隙至少为$\frac{16}{7}$，这改进了数十年来保持为2的下界结果。更重要的是，我们发展了证明此类下界的新技术，这些技术在其他场景中也可能具有应用价值。