The graph invariant EPT-sum has cropped up in several unrelated fields in later years: As an objective function for hierarchical clustering, as a more fine-grained version of the classical edge ranking problem, and, specifically when the input is a vertex-weighted tree, as a measure of average/expected search length in a partially ordered set. The EPT-sum of a graph $G$ is defined as the minimum sum of the depth of every leaf in an edge partition tree (EPT), a rooted tree where leaves correspond to vertices in $G$ and internal nodes correspond to edges in $G$. A simple algorithm that approximates EPT-sum on trees is given by recursively choosing the most balanced edge in the input tree $G$ to build an EPT of $G$. Due to its fast runtime, this balanced cut algorithm is used in practice. In this paper, we show that the balanced cut algorithm gives a 1.5-approximation of EPT-sum on trees, which amounts to a tight analysis and answers a question posed by Cicalese et al. in 2014.

翻译：图不变量EPT-sum近年来在多个不相关领域中涌现：作为层次聚类的目标函数，作为经典边排序问题的更细粒度版本，以及特别地，当输入为顶点加权树时，作为偏序集中平均/期望搜索长度的度量。图$G$的EPT-sum定义为边划分树（一种以$G$中顶点为叶子、以$G$中边为内部节点的有根树）中所有叶子深度之和的最小值。一种近似树中EPT-sum的简单算法通过递归选择输入树$G$中最平衡的边来构建其EPT。由于运行速度快，这种平衡切割算法在实际中得到了应用。本文证明，平衡切割算法在树上给出了EPT-sum的1.5-近似，这构成了紧分析，并回答了Cicalese等人于2014年提出的问题。