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。给定一个终端集合 $T$ 和一个以 $T$ 的真子集为索引的 $(2^{|T|}-2)$ 维向量 $\pi$，是否存在一个包含 $T$ 的图 $G$，使得对所有满足 $\emptyset\subsetneq S\subsetneq T$ 的子集 $S$，$\pi_S$ 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 给出的特殊线性不等式类。本文的主要结果是关于层状族的一类新线性不等式，它推广了之前的所有结果。利用新不等式类，我们可将 Karger 的近似最小割计数结果推广至含终端的图。