The Complexity of Bicriteria Tree-Depth

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, mainly due to potential applications. These include purely combinatorial applications like searching in tree-like partial orders (which generalizes binary search in sorted data), or practical ones in parallel processing.