The tree-depth problem can be seen as finding an elimination tree of minimum height for a given input graph $G$. We introduce a bicriteria generalization in which additionally the width of the elimination tree needs to be bounded by some input integer $b$. We are interested in the case when $G$ is the line graph of a tree, proving that the problem is NP-hard and obtaining a polynomial-time additive $2b$-approximation algorithm. This particular class of graphs received significant attention in the past, mainly due to a number of potential applications, e.g. in parallel assembly of modular products, or parallel query processing in relational databases, as well as purely combinatorial applications, including searching in tree-like partial orders (which in turn generalizes binary search on sorted data).

翻译：树深度问题可以被视为为某一输入图找到最低高度的树。我们采用了双标准的一般化,除掉树的宽度还需要受某些输入整数美元的约束。我们感兴趣的是,$G是树的线形图,这证明问题在于NP-硬,并获得一个多米-时间添加2b$-协调算法。这一特殊类别的图表过去受到极大关注,这主要是因为一些潜在的应用,例如模块产品平行组装,或相关数据库的平行查询处理,以及纯粹的组合应用程序,包括类树类部分订单的搜索(这反过来又概括了分类数据的二元搜索)。