Gomory-Hu子树何时存在？
When Do Gomory-Hu Subtrees Exist?
摘要：
Gomory-Hu（GH）树是一种经典的图连通性稀疏化技术。它是组合优化的基本模型之一，也不断发现新的应用，最近在社会网络分析。对于任何边容量限制的无向图$G=(V,E)$和{ \em终端} $Z \subseteq V$的任何子集，Gomory-Hu树是边容量限制的树$T=(Z,E(T))$，使得对于每个$u,v \in Z$，$G$中的最小容量$uv$割的值与$T$中的相同。此外，$T$中的最小切割直接识别（以某种方式）$G$中的最小切割。众所周知，我们并不总是能找到一棵GH树，它是$G$的一个子图。例如，$K_{3,3}$顶点的每棵GH树都是$5$ -星。我们描述了那些图和终端对$(G,Z)$，总是承认这样的树。我们证明了这些图没有\emph{terminal-$K_{2,3}$ minor}。也就是说，没有顶点对应于$Z$中终端的$K_{2,3}$子节点。我们还表明，家庭对$(G,Z)$禁止这样的$K_{2,3}$“$Z$ -未成年人”产生，粗略地说，从所谓的Okamura-西摩的实例。更准确地说，它们是{ \em $Z$ -webs}的子图。一个$Z$ -网络是建立在平面图与一个外面，其中包含所有的终端和每个内面是一个三角形，其中可以包含一个任意的图。这种特性产生了一个额外的后果多流问题。修复图形$G$和端子的子集$Z \subseteq V(G)$。调用$(G,Z)$ { \em cut-sufficient}，如果切割条件足以表征$Z$中顶点之间的任何需求以及$G$上的任何边容量的多流的存在性。则$(G,Z)$是充分切割的，当且仅当它是无终端$K_{2,3}$的。
英文摘要：
Gomory-Hu (GH) Trees are a classical sparsification technique for graph connectivity. It is one of the fundamental models in combinatorial optimization which also continually finds new applications, most recently in social network analysis. For any edge-capacitated undirected graph $G=(V,E)$ and any subset of {\em terminals} $Z \subseteq V$, a Gomory-Hu Tree is an edge-capacitated tree $T=(Z,E(T))$ such that for every $u,v \in Z$, the value of the minimum capacity $uv$ cut in $G$ is the same as in $T$. Moreover, the minimum cuts in $T$ directly identify (in a certain way) those in $G$. It is well-known that we may not always find a GH tree which is a subgraph of $G$. For instance, every GH tree for the vertices of $K_{3,3}$ is a $5$-star. We characterize those graph and terminal pairs $(G,Z)$ which always admit such a tree. We show that these are the graphs which have no \emph{terminal-$K_{2,3}$ minor}. That is, no $K_{2,3}$ minor whose vertices correspond to terminals in $Z$. We also show that the family of pairs $(G,Z)$ which forbid such $K_{2,3}$ "$Z$-minors" arises, roughly speaking, from so-called Okamura-Seymour instances. More precisely, they are subgraphs of {\em $Z$-webs}. A $Z$-web is built from planar graphs with one outside face which contains all the terminals and each inner face is a triangle which may contain an arbitrary graph. This characterization yields an additional consequence for multiflow problems. Fix a graph $G$ and a subset $Z \subseteq V(G)$ of terminals. Call $(G,Z)$ {\em cut-sufficient} if the cut condition is sufficient to characterize the existence of a multiflow for any demands between vertices in $Z$, and any edge capacities on $G$. Then $(G,Z)$ is cut-sufficient if and only if it is terminal-$K_{2,3}$ free.
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