Weighted Treedepth is NP-complete on Graphs of Bounded Degree

2025-10-21

AI Summary

This paper investigates the computational complexity of weighted treedepth on bounded-degree graphs. Specifically, it addresses the decision problem: given a vertex-weighted graph of bounded maximum degree, does there exist a treedepth decomposition whose total weight (sum of node weights in the decomposition) is at most a given threshold? The authors establish, for the first time, that this problem is NP-complete—via a polynomial-time reduction from 3-SAT, carefully engineered to preserve bounded maximum degree in the resulting instance. Furthermore, they identify tractable cases by showing the problem is solvable in polynomial time on paths and 1-subdivided stars, thereby delineating precise complexity boundaries. These results fill a fundamental gap in the complexity landscape of weighted structural graph parameters on sparse graphs and provide essential theoretical foundations for designing algorithms based on treedepth.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Models for Web evolution

Abstract

A treedepth decomposition of an undirected graph $G$ is a rooted forest $F$ on the vertex set of $G$ such that every edge $uvin E(G)$ is in ancestor-descendant relationship in $F$. Given a weight function $wcolon V(G) ightarrow mathbb{N}$, the weighted depth of a treedepth decomposition is the maximum weight of any path from the root to a leaf, where the weight of a path is the sum of the weights of its vertices. It is known that deciding weighted treedepth is NP-complete even on trees. We prove that weighted treedepth is also NP-complete on bounded degree graphs. On the positive side, we prove that the problem is efficiently solvable on paths and on 1-subdivided stars.

Problem

• Weighted treedepth is NP-complete for bounded degree graphs
• The problem extends NP-completeness from trees to bounded degree graphs
• It provides efficient solutions for paths and 1-subdivided stars

Innovation

• Proves NP-completeness on bounded degree graphs
• Solves weighted treedepth efficiently on paths
• Provides efficient solution for 1-subdivided stars