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

TL;DR

This work analyzes MinHLT, the problem of determining whether a graph admits a DFS tree of height at most $k$, situating it within bounded treedepth decompositions. It delivers an explicit FPT algorithm for MinHLT, first achieving $2^{\mathcal{O}(k \log k)} \cdot n$ time and then an optimized $k^{\mathcal{O}(k)} \cdot n$ variant via dynamic programming on treedepth-structured decompositions. The authors also establish that MinHLT is NP-hard on chordal graphs, while the dual problem DualMinHLT is FPT on chordal graphs; more generally, DualMinHLT, MinLLT, and MaxLLT are FPT parameterized by $k$ plus treewidth through MSO$_2$ and Courcelle’s theorem. Additionally, an MSO$_2$ formulation for these problems on general graphs provides a unified framework for fixed-parameter tractability results. Overall, the paper advances understanding of bounded-height DFS trees and their algorithmic implications, while highlighting open questions about the exact parameterized complexity of DualMinHLT.

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, Sánchez 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.

Theorems & Definitions (40)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 3.1
• Theorem 3.2: Dereniowski and Nadolski ipl2006
• Lemma 3.3: Dereniowski and Nadolski ipl2006
• proof