[论文解读] The First Known Problem That Is FPT with Respect to Node Scanwidth but Not Treewidth

Jannik Schestag, Norbert Zeh|arXiv (Cornell University)|Feb 6, 2026

Genomics and Phylogenetic Studies被引用 0

一句话总结

论文证明了复杂度分离：带依赖的加权系统性多样性在节点扫描宽度下是FPT参数化的，但在树宽度下对所有ℓ≥1都是W[ℓ]-hard。

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.

研究动机与目标

• 研究DAGs中以扫描宽度作为宽度度量的动机，应用于系统发育网络。
• 证明在某些情况下扫描宽度能产生FPT算法，而树宽度则不行。
• 建立一个自然问题在节点扫描宽度与树宽度之间的 hardness 分离。
• 在食物网（有向无环图）上对带依赖的加权系统性多样性（Weighted PDD_s）建模与分析。
• 通过精确的归约提供扫描宽度与树宽度的分离构造。

提出的方法

• 在带顶点权重d、弧权重γ和整数B、D的食物网（有向无环图）上定义 Weighted PDD_s。
• 通过从 Capacitated Dominating Set 的归约，证明1/2-PDD_s 相对于树宽度的W[ℓ]-hard性。
• 构造一个多项式时间可减的Weighted PDD_s实例，使树宽度得到受控且保持实例等价。
• 引入并分析选择器小部件和配额小部件，在 CDS → 1/2-PDD_s 的归约中强制支配集合与预算约束。
• 开发基于树扩展T的动态规划方法，以节点扫描宽度参数化求解Weighted PDD_s，使用特化的(v,A′,ℓ)-兼容性状态。
• 给定节点扫描宽度 NSW(F) 的树扩展时，证明解可在时间 O(2^{nsw(F)} · n^3) 内获得。

实验结果

研究问题

• RQ1 是否存在对节点扫描宽度FPT且对树宽度非FPT的问题？
• RQ2 对于与系统发育网络相关的自然问题，是否存在扫描宽度与树宽度之间的强分离？
• RQ3 尽管在树宽度下困难，Weighted PDD_s 能否在节点扫描宽度参数化下高效求解？
• RQ4 选择器小部件与配额小部件如何促进从 CDS 到1/2-PDD_s 的归约？
• RQ5 树扩展、节点扫描宽度与Weighted PDD_s 的DP公式之间的精确关系是什么？

主要发现

• 在树宽度参数化下，1/2-PDD_s 对所有ℓ≥1 都是 W[ℓ]-hard。
• 在给定树扩展的 NSW(F) 下，Weighted PDD_s 相对于节点扫描宽度是FPT，时间复杂度为 O(2^{nsw(F)} · n^3)。
• Capacitated Dominating Set（CDS）实例可以在多项式时间内被化归为1/2-PDD_s，同时保持树宽度至常数因子内的等价性。
• 归约使用选择器小部件和配额小部件来在构造的食物网中强制支配集结构与容量约束。
• 基于节点扫描宽度的DP 利用类似树分解的结构 T，在子树间管理兼容状态。