Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jannik Schestag, Norbert Zeh

TL;DR

It is proved 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 $\ell \ge 1$.

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.

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

TL;DR

It is proved that Weighted Phylogenetic Diversity with Dependencies is FPT with respect to the scanwidth of the food web but W[]-hard with respect to its treewidth, for all .

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[]-hard, for some , 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[]-hard with respect to its treewidth, for all . To the best of our knowledge, no such separation between these two width measures has been shown for any problem before.
Paper Structure (7 sections, 6 theorems, 5 equations, 3 figures)

This paper contains 7 sections, 6 theorems, 5 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $1/2$-PDD$_{\text{s}}$ Parameterized by Treewidth
  4. Weighted PDD$_{\text{s}}$ Parameterized by Node Scanwidth
  5. Leaf
  6. Internal Node
  7. Discussion

Key Result

Theorem 1

$1/2$-PDD$_{\text{s}}$ is ${\normalfont W[\ell]}\xspace$-hard, for all $\ell \ge 1$, when parameterized by the treewidth of the food web.

Figures (3)

  • Figure 1: The reduction from CDS to $1/2$-PDD$_{\text{s}}$. (a) An input graph $G$ of CDS. The capacities are shown beside each vertex. Solid vertices are in a dominating set $S$; hollow vertices are not. Arrows on edges represent the mapping $f$ from vertices in $V \setminus S$ to $S$. (b) The food web $\mathcal{F}$ constructed from $G$. Only the root of each widget is shown. Solid vertices are in a viable set $S'$ corresponding to $(S,f)$. Hollow vertices are not.
  • Figure 2: A selector widget $S_v$. Dashed arcs indicate predators of $v_0$ outside of $S_v$. There are no other arcs to or from vertices outside of $S_v$.
  • Figure 3: A quota widget $Q$ of type $Q(\delta,\ell,r,M)$ and with root $x$. Dashed arcs indicate connections to neighbours of $x$ outside of $Q$. There are no other arcs to or from vertices outside of $Q$.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9