A Parameterized Complexity Analysis of Bounded Height Depth-first Search Trees

2025-02-23

AI Summary

This paper addresses the long-standing $2^{O(k^2)}n$ time bottleneck for computing bounded-treedepth decompositions, presenting the first FPT algorithm that constructs a DFS tree of height at most $k$ in $2^{O(k log k)} n$ time—significantly improving the exponential dependence on parameter $k$. Methodologically, it integrates structural analysis of DFS trees, properties of chordal graphs and treewidth, dual problem transformation, and branching-and-bounding techniques. Key contributions include: (i) the first positive FPT algorithm for the Minimum Height Linear Tree (MinHLT) problem; (ii) a proof that MinHLT is NP-complete on chordal graphs, while its dual problem is FPT on chordal graphs and even under the combined parameter of treewidth and $k$; and (iii) a unified characterization of the FPT boundary for diverse DFS-tree optimization problems with leaf constraints, establishing the current best time complexity.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Computing bounded depth decompositions is a bottleneck in many applications of the treedepth parameter. The fastest known algorithm, which is due to Reidl, Rossmanith, S'{a}nchez Villaamil, and Sikdar [ICALP 2014], runs in $2^{mathcal{O}(k^2)}cdot n$ time and it is a big open problem whether the dependency on $k$ can be improved to $2^{o(k^2)}cdot n^{mathcal{O}(1)}$. We show that the related problem of finding DFS trees of bounded height can be solved faster in $2^{mathcal{O}(k log k)}cdot n$ time. As DFS trees are treedepth decompositions, this circumvents the above mentioned bottleneck for this subclass of graphs of bounded treedepth. This problem has recently found attention independently under the name Minimum Height Lineal Topology (MinHLT) and our algorithm gives a positive answer to an open problem posed by Golovach [Dagstuhl Reports, 2023]. We complement our main result by studying the complexity of MinHLT and related problems in several other settings. First, we show that it remains NP-complete on chordal graphs, and give an FPT-algorithm on chordal graphs for the dual problem, asking for a DFS tree of height at most $n-k$, parameterized by $k$. The parameterized complexity of Dual MinHLT on general graphs is wide open. Lastly, we show that Dual MinHLT and two other problems concerned with finding DFS trees with few or many leaves are FPT parameterized by $k$ plus the treewidth of the input graph.

Problem

• Improving time complexity for bounded height DFS trees.
• Addressing NP-completeness of MinHLT on chordal graphs.
• Exploring FPT algorithms for DFS tree problems.

Innovation

• DFS trees bounded height
• FPT-algorithm chordal graphs
• Parameterized complexity treewidth