The First Known Problem That Is FPT with Respect to Node Scanwidth but Not Treewidth

2026-02-06

AI Summary

This study investigates the fundamental differences in parameterized complexity between the graph width parameters scanwidth and treewidth, specifically addressing whether there exist problems that are fixed-parameter tractable (FPT) with respect to scanwidth but intractable with respect to treewidth. By leveraging structural properties of directed acyclic graphs and introducing a direction-respecting variant of scanwidth, the authors combine techniques from parameterized complexity theory, structural graph analysis, and topological ordering to establish that the Weighted Phylogenetic Diversity with Dependencies problem is FPT when parameterized by scanwidth, yet W[ℓ]-hard for all ℓ ≥ 1 when parameterized by treewidth. This result provides the first known complexity separation between these two width measures, demonstrating that scanwidth captures structural information beyond the expressive power of treewidth.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Structural parameters of graphs, such as treewidth, play a central role in the study of the parameterized complexity of graph problems. Motivated by the study of parametrized algorithms on phylogenetic networks, scanwidth was introduced recently as a new treewidth-like structural parameter for directed acyclic graphs (DAGs) that respects the edge directions in the DAG. The utility of this width measure has been demonstrated by results that show that a number of problems that are fixed-parameter tractable (FPT) with respect to both treewidth and scanwidth allow algorithms with a better dependence on scanwidth than on treewidth. More importantly, these scanwidth-based algorithms are often much simpler than their treewidth-based counterparts: the name ``scanwidth''reflects that traversing a tree extension (the scanwidth-equivalent of a tree decomposition) of a DAG amounts to ``scanning''the DAG according to a well-chosen topological ordering. While these results show that scanwidth is useful especially for solving problems on phylogenetic networks, all problems studied through the lens of scanwidth so far are either FPT with respect to both scanwidth and treewidth, or W[$\ell$]-hard, for some $\ell \ge 1$, with respect to both. In this paper, we show that scanwidth is not just a proxy for treewidth and provides information about the structure of the input graph not provided by treewidth, by proving a fairly stark complexity-theoretic separation between these two width measures. Specifically, we prove that Weighted Phylogenetic Diversity with Dependencies is FPT with respect to the scanwidth of the food web but W[$\ell$]-hard with respect to its treewidth, for all $\ell \ge 1$. To the best of our knowledge, no such separation between these two width measures has been shown for any problem before.

Problem

• scanwidth
• treewidth
• fixed-parameter tractability
• phylogenetic networks
• parameterized complexity

Innovation

• scanwidth
• treewidth
• fixed-parameter tractability
• parameterized complexity
• phylogenetic networks