Pushing the frontiers of subexponential FPT time for Feedback Vertex Set

TL;DR

This work develops a general framework of nice hereditary graph classes with bounded tree neighborhood complexity to obtain subexponential FPT algorithms for Feedback Vertex Set parameterized by $k$. The approach combines kernelization, partitioning into $t$-uniform components, and tilde-$K_{t,t}$-driven branching with a treewidth-based DP on bounded-core instances, yielding running times of the form $2^{O(k^{ u})}n^{O(1)}$ for some $ u<1$ depending on the class. It unifies and extends subexponential results across planar, map, unit-disk, pseudo-disk, and bounded-degree string graphs, and applies to new geometric classes such as segment graphs and $s$-string graphs, with robust (representation-free) running times. The paper also provides concrete parameterizations for pseudo-disk and $s$-string graphs and discusses limitations and directions for extending the framework to higher dimensions or classes excluding induced minors.

Abstract

The paper deals with the Feedback Vertex Set problem parameterized by the solution size. Given a graph $G$ and a parameter $k$, one has to decide if there is a set $S$ of at most $k$ vertices such that $G-S$ is acyclic. Assuming the Exponential Time Hypothesis, it is known that FVS cannot be solved in time $2^{o(k)}n^{\mathcal{O}(1)}$ in general graphs. To overcome this, many recent results considered FVS restricted to particular intersection graph classes and provided such $2^{o(k)}n^{\mathcal{O}(1)}$ algorithms. In this paper we provide generic conditions on a graph class for the existence of an algorithm solving FVS in subexponential FPT time, i.e. time $2^{k^\varepsilon} \mathop{\rm poly}(n)$, for some $\varepsilon<1$, where $n$ denotes the number of vertices of the instance and $k$ the parameter. On the one hand this result unifies algorithms that have been proposed over the years for several graph classes such as planar graphs, map graphs, unit-disk graphs, pseudo-disk graphs, and string graphs of bounded edge-degree. On the other hand it extends the tractability horizon of FVS to new classes that are not amenable to previously used techniques, in particular intersection graphs of ``thin'' objects like segment graphs or more generally $s$-string graphs.

Figures (7)

• Figure 1: Example of a $K_{r,r}$ contained as a minor for $r=4$ in a disk graph (left) and a $2$-DIR graph (right). In the case of disk graph, $v$ has a matching of size $r-2$ in its neighborhood, forming a triangle bundle, which can be exploited to branch. The set $M$ are depicted in blue. For the $2$-DIR graph, the vertices of the long paths are represented by segments with small variation in their height and not intersecting for better clarity, but are in fact on the same level and intersecting.
• Figure 2: Example of partition where $P_i$ is partitioned into $x(P_i)=3$ subpaths, with $P_i^1$ and $P_i^2$ in $\mathcal{T}^+$ and $P_i^3 \in \mathcal{T}^-$.
• Figure 3: Example of $\widetilde{K_{t,t}}$ for $t=4$. Here we have $X\subseteq N_M(T_i)$ for $1\leq i \leq 4$.
• Figure 4: Representation of a $t$-uniform partition of a subtree $T$ of $G-M$. Here are represented only vertices of $F$ (and not vertices of $M$). Vertices of the trees in $\mathcal{T}^+$ (respectively $\mathcal{T}^-$) are represented in red (respectively blue), so as the edges between two vertices in the same tree of $\mathcal{T}^+$ (respectively $\mathcal{T}^-$). Edges between distinct trees of $\mathcal{T}$ are represented in black.
• Figure 5: Transformation of a union of $s$-strings to a region homeomorphic to a disk, while keeping the same neighborhood in $A$. The strings of $A$ are depicted in black.

Theorems & Definitions (79)

• Theorem 1.2: lee2016separators
• Theorem 1.3: lee2016separatorsDVORAK2019137
• Definition 1.4
• Definition 1.5
• Corollary 1.5
• Corollary 1.5
• Theorem 3.1: 2approx-fvs-12approx-fvs-2
• Lemma 3.2
• proof
• Lemma 3.3