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

TL;DR

This paper demonstrates a fundamental difference between scanwidth and treewidth by showing a problem that is fixed-parameter tractable with respect to scanwidth but W[\ell]-hard with respect to treewidth, revealing that scanwidth captures unique structural graph properties.

Contribution

The paper provides the first known example of a problem that is FPT with respect to scanwidth but hard for treewidth, establishing a significant separation between these two graph parameters.

Findings

Weighted Phylogenetic Diversity with Dependencies is FPT in scanwidth.

The same problem is W[\ell]-hard in treewidth for all \ell \ge 1.

Scanwidth captures structural graph information beyond treewidth.

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.