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 can be used in practice, and has earlier been analysed to give a 1.62-approximation on trees. 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：近年来，图不变量EPT-sum在多个不相关领域中出现：作为层次聚类的目标函数，作为经典边排序问题的细粒度版本，特别当输入为顶点加权树时，可作为偏序集中平均/期望搜索长度的度量。图$G$的EPT-sum定义为边划分树（EPT）中所有叶节点深度之和的最小值，其中EPT是一种有根树，其叶节点对应$G$中的顶点，内部节点对应$G$中的边。一种近似计算树上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$G$的EPT。由于其快速运行时间，该平衡割算法可用于实际场景，且先前分析表明其在树上具有1.62近似比。本文证明平衡割算法对树上EPT-sum具有1.5近似比，这构成了紧分析并回答了Cicalese等人于2014年提出的问题。