We study the following characterization problem. Given a set $T$ of terminals and a $(2^{|T|}-2)$-dimensional vector $\pi$ whose coordinates are indexed by proper subsets of $T$, is there a graph $G$ that contains $T$, such that for all subsets $\emptyset\subsetneq S\subsetneq T$, $\pi_S$ equals the value of the min-cut in $G$ separating $S$ from $T\setminus S$? The only known necessary conditions are submodularity and a special class of linear inequalities given by Chaudhuri, Subrahmanyam, Wagner and Zaroliagis. Our main result is a new class of linear inequalities concerning laminar families, that generalize all previous ones. Using our new class of inequalities, we can generalize Karger's approximate min-cut counting result to graphs with terminals.

翻译：我们研究以下刻画问题：给定一个终端集$T$和一个维度为$(2^{|T|}-2)$的向量$\pi$，其坐标由$T$的真子集索引，是否存在一个包含$T$的图$G$，使得对于所有子集$\emptyset\subsetneq S\subsetneq T$，$\pi_S$等于$G$中分离$S$与$T\setminus S$的最小割的值？目前已知的必要条件仅包括子模性以及由Chaudhuri、Subrahmanyam、Wagner和Zaroliagis给出的一类特殊线性不等式。我们的主要结果是关于层状族的一类新线性不等式，它概括了所有先前的不等式。利用这类新不等式，我们可将Karger的近似最小割计数结果推广到带终端的图上。