Losing Treewidth In The Presence Of Weights

TL;DR

The presented algorithm is based on a novel technique of random sampling of so-called protrusions and admits a randomized polynomial-time constant-factor approximation algorithm for every fixed $\eta$.

Abstract

In the Weighted Treewidth-$η$ Deletion problem we are given a node-weighted graph $G$ and we look for a vertex subset $X$ of minimum weight such that the treewidth of $G-X$ is at most $η$. We show that Weighted Treewidth-$η$ Deletion admits a randomized polynomial-time constant-factor approximation algorithm for every fixed $η$. Our algorithm also works for the more general Weighted Planar $F$-M-Deletion problem. This work extends the results for unweighted graphs by [Fomin, Lokshtanov, Misra, Saurabh; FOCS '12] and answers a question posed by [Agrawal, Lokshtanov, Misra, Saurabh, Zehavi; APPROX/RANDOM '18] and [Kim, Lee, Thilikos; APPROX/RANDOM '21]. The presented algorithm is based on a novel technique of random sampling of so-called protrusions.

Figures (2)

• Figure 1: The set $X$ is a treewidth-$\eta$ modulator and $N(X)$ is represented by black discs. The blue sets $S_1, S_2$ are two bags in a tree decomposition of $G-X$ that contain the marked vertices. The yellow and orange sets are the connected components of $G - (X \cup S_1 \cup S_2)$. For $i = 1,2,3$ we have that $G[C_i]$ is connected, $\mathrm{\textbf{tw}}(G[C_i]) \le \eta$, and $N(C_i) = S_2$, hence $(C_i,S_2)$ is a simple $\eta$-separation. The pair $(C_1 \cup C_2 \cup C_3,\, S_2)$ forms a semi-simple $\eta$-separation. The vertex set $C_4$ is enclosed by the dashed cycle. The simple $\eta$-separation $(C_4, N(C_4))$ covers two edges outgoing from $X$ and so the set $N(C_4) \setminus X$ has been marked. In the proof we take advantage of the fact that there can be no inclusion-wise maximal simple $\eta$-separation $(C,S)$ for which $C$ is contained in the yellow subset of $C_4$.
• Figure 2: An illustration of the case analysis in \ref{['lem:protr:cover']}. The graph $G-X$ has treewidth bounded by $\eta$ and $\widehat{D} \subseteq V(G) \setminus X$ is the union of vertices from the marked bags, drawn in blue. The light-gray polygons are the connected components of $G - (X \cup \widehat{D})$. Each orange set represents $C_{sim}$ in some scenario. The options 1 and 2 illustrate the Case (1.a.i). If $C_{sim}$ is a proper subset of $\widehat{C}$ (the gray one in option 1) then $(C_{sim},S)$ is not $\vartriangleleft$-maximal. If $C_{sim} = \widehat{C}$ then $S$ is located within two bags of the tree decomposition of $G-X$ and this option boils down to \ref{['lem:protr:lca']}. The option 3 corresponds to Case (1.a.ii). Here the dashed rectangle depicts the interior of a $\vartriangleleft$-greater simple $\eta$-separation (denoted $(C^v_{sim},S_v)$ in the proof) which covers the three vertices from $N(X)$ as well as $C_{sim}$. Also, two arrangements of the edge $xy$ from Case (2.a) are showed in purple. The option 4 illustrates Case (1.b) with $v \in C_{sim} \cap \widehat{D}$. By \ref{['lem:protr:important']} the set $S = N(C_{sim})$ must be an important $(v,X)$-separator.

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 3.1: cygan2015parameterized
• Lemma 3.2: cygan2015parameterized
• Definition 3.3
• Definition 3.4
• Lemma 3.5: FominLMS12
• Theorem 3.6: Bodlaender96
• Definition 3.7