The Complexity of Bicriteria Tree-Depth

TL;DR

This paper studies a bicriteria version of the tree-depth problem, proving NP-hardness for line graphs of trees and providing a polynomial-time approximation algorithm, with implications for various applications in parallel processing and combinatorics.

Contribution

It introduces a bicriteria generalization of tree-depth, proves NP-hardness for a specific class of graphs, and offers a polynomial-time approximation algorithm.

Findings

NP-hardness for line graphs of trees

Polynomial-time additive 2b-approximation algorithm

Relevance to applications in parallel processing and combinatorics

Abstract

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).