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.

翻译：图的结构参数（如树宽度）在图问题的参数化复杂度研究中占据核心地位。受系统发育网络参数化算法研究的推动，扫描宽度作为一类新的类树宽度结构参数被提出，适用于有向无环图（DAG），且该参数遵循DAG的边方向。已有研究证明该宽度度量具有实用价值：一系列关于树宽度和扫描宽度均固定参数可解（FPT）的问题，其算法对扫描宽度的依赖度优于对树宽度的依赖度。更重要的是，基于扫描宽度的算法通常比基于树宽度的对应算法更为简洁：“扫描宽度”这一名称反映了遍历DAG的树扩展（即扫描宽度对应的树分解）相当于按照精心选择的拓扑序对DAG进行“扫描”。尽管这些结果表明扫描宽度尤其适用于解决系统发育网络问题，但迄今为止通过扫描宽度研究的所有问题要么对扫描宽度和树宽度均为FPT，要么对两者均为W[ℓ]困难（其中ℓ ≥ 1）。本文通过证明这两种宽度度量之间存在显著的计算复杂度分离，表明扫描宽度不仅是树宽度的替代指标，更能提供树宽度无法揭示的输入图结构信息。具体而言，我们证明了带依赖关系的加权系统发育多样性问题对于食物网的扫描宽度是FPT的，而对于其树宽度是W[ℓ]困难的（对所有ℓ ≥ 1均成立）。据我们所知，此前尚未有任何问题在这两种宽度度量间展现出此类分离特性。