Tight Bounds for Chordal/Interval Vertex Deletion Parameterized by Treewidth

TL;DR

This work analyzes chordal and interval vertex deletion under the treewidth parameter. It introduces a novel matroid-based connection between chordality and graphic matroids, enabling representative-family techniques to drive a single-exponential parameterized algorithm for Chordal Vertex Deletion with running time $2^{O(\mathrm{tw})}\cdot n$ and base $c=2^{\omega-1}\cdot 3 + 1$, while establishing an ETH-based lower bound that Interval Vertex Deletion cannot be improved to $2^{o(\mathrm{tw}\log \mathrm{tw})}\cdot n$. The interval deletion lower bound is proved via a reduction from $k\times k$ Permutation Clique, using permutation and choice gadgets to encode the problem within interval graphs and treedepth bounds. The paper thus achieves a sharp separation between the two problems under tw, improving the ChVD frontier and tightly bounding Interval Vertex Deletion, and it introduces a matroid-representative-families framework that may extend to other connectivity/deletion problems on bounded-treewidth graphs.

Abstract

In Chordal/Interval Vertex Deletion we ask how many vertices one needs to remove from a graph to make it chordal (respectively: interval). We study these problems under the parameterization by treewidth $tw$ of the input graph $G$. On the one hand, we present an algorithm for Chordal Vertex Deletion with running time $2^{O(tw)} \cdot |V(G)|$, improving upon the running time $2^{O(tw^2)} \cdot |V(G)|^{O(1)}$ by Jansen, de Kroon, and Wlodarczyk (STOC'21). When a tree decomposition of width $tw$ is given, then the base of the exponent equals $2^{ω-1}\cdot 3 + 1$. Our algorithm is based on a novel link between chordal graphs and graphic matroids, which allows us to employ the framework of representative families. On the other hand, we prove that the known $2^{O(tw \log tw)} \cdot |V(G)|$-time algorithm for Interval Vertex Deletion cannot be improved assuming Exponential Time Hypothesis.

Figures (3)

• Figure 1: On the left: graph $G$ and set $X \subseteq V(G)$ represented by black disks. The graph $G[X]$ is drawn with solid edges. There are two minimal vertex separators in $G[X]$: $S_1 = \{v\}$ and $S_2 = \{u,v\}$, sketched in gray. In the middle: the graph $\texttt{Aux}(G,X,S_1)$ with thick edges indicating a component that gets contracted into a single vertex; the gray vertices and edges are removed. On the right: the graph $\texttt{Aux}(G,X,S_2)$; note that $|\texttt{Comp}(G[X],S_2)|=2$ because the lower vertices of $X$ are not adjacent to every vertex in $S_2$. The graph $\texttt{Aux}(G,X,S_1)$ contains a cycle and this witnesses that $G$ is not chordal. However, removing from $G$ any single vertex among $x,y,z$ results in a chordal graph.
• Figure 2: Illustration for \ref{['lem:interval:permutation']}. The intervals for vertices of $Y_4$ are blank, ordered from bottom to top. They encode permutation $(2,4,3,1)$. The black intervals represent vertices $x_1,x_2,x_3,x_4,x_5$ with neighborhoods encoding sets $\{2\}$, $\{2,4\}$ (twice), $\{2,4,3\}$, and $\{2,4,3,1\}$.
• Figure 3: Top: the choice gadget $H_5$ with the subgraph $Q_1$ highlighted in green. The copies of $P$ are sketched symbolically with dashed lines and the squares represent vertices $g^\alpha_i$. The red disks and squares represent a solution constructed in \ref{['lem:interval:choice-properties']}(2). This solution 'chooses' $i=2$, leaves untouched the four vertices $g^\alpha_2$, and removes $h^\alpha_2$ as well as $g^\alpha_i$ for $i \ne 2$. Bottom left: the graph $P$ and vertices named $h,g$. Two vertex-disjoint non-interval subgraphs of $P$ have green edges. Bottom right: a closer look at the first two blocks of $H_5$ with two copies of $P$ drawn in detail. The subgraph highlighted in green witnesses that if a minimum-size solution removes $g^\alpha_i$ for at least one $\alpha \in [4]$ then it must also remove $v^2_i$, what is exploited in \ref{['lem:interval:choice-properties']}(3).

Theorems & Definitions (57)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4: $\star$
• Lemma 5: $\star$
• Lemma 7: BrandstadtLS99
• Lemma 8: BrandstadtLS99
• Definition 10: Treewidth
• Definition 11
• Definition 12: Treedepth