Data reduction for directed feedback vertex set on graphs without long induced cycles

TL;DR

This work investigates kernelization for the Directed Feedback Vertex Set (DFVS) problem on directed graphs without long induced cycles. It establishes a $2^d k^d$-vertex kernel with a bound of $d^{3d} k^d$ on the number of short cycles, and proves a W[1]-hardness barrier for efficiently identifying vertices on induced cycles of length at most $d$, motivating implicit-structure kernelization. The authors extend results to sparse graph classes, delivering kernels of size $f_{\\mathscr{C}}(d)\cdot k$ for bounded expansion and $f_{\mathscr{C}}(d,\varepsilon)\cdot k^{1+\varepsilon}$ for nowhere-dense classes, and show planar graphs yield treewidth $O(d)$ permitting $2^{O(d)}$-time DFVS solutions. They introduce a new reduction rule that subsumes prior rules and enables a kernel with respect to the underlying undirected FVS size $f$, achieving $O(f^3 k)$ vertices, complemented by a detailed size analysis. Finally, the paper offers an LP-based approximation via a polynomial-size order-ILP with a constant-factor guarantee relative to the cycles-ILP and proves hardness results for directed chordless paths, informing the boundaries of reduction-based approaches.

Abstract

We study reduction rules for Directed Feedback Vertex Set (DFVS) on directed graphs without long cycles. A DFVS instance without cycles longer than $d$ naturally corresponds to an instance of $d$-Hitting Set, however, enumerating all cycles in an $n$-vertex graph and then kernelizing the resulting $d$-Hitting Set instance can be too costly, as already enumerating all cycles can take time $Ω(n^d)$. We show how to compute a kernel with at most $2^dk^d$ vertices and at most $d^{3d}k^d$ induced cycles of length at most $d$, where $k$ is the size of a minimum directed feedback vertex set. We then study classes of graphs whose underlying undirected graphs have bounded expansion or are nowhere dense. We prove that for every nowhere dense class $\mathscr{C}$ there is a function $f_\mathscr{C}(d,ε)$ such that for graphs $G\in \mathscr{C}$ without induced cycles of length greater than $d$ we can compute a kernel with $f_\mathscr{C}(d,ε)\cdot k^{1+ε}$ vertices for any $ε>0$ in time $f_\mathscr{C}(d,ε)\cdot n^{O(1)}$. The most restricted classes we consider are strongly connected planar graphs without any (induced or non-induced) long cycles. We show that these classes have treewidth $O(d)$ and hence DFVS on planar graphs without cycles of length greater than $d$ can be solved in time $2^{O(d)}\cdot n^{O(1)}$. We finally present a new data reduction rule for general DFVS and prove that the rule together with two standard rules subsumes all rules applied in the work of Bergougnoux et al.\ to obtain a polynomial kernel for DFVS[FVS], i.e., DFVS parameterized by the feedback vertex set number of the underlying (undirected) graph. We conclude by studying the LP-based approximation of DFVS.

Figures (4)

• Figure 1: Illustration of a cycle induced around a vertex $v$. Dotted red edges are forbidden.
• Figure 2: Illustration of \ref{['lem:non-edge-contributions']}.
• Figure 3: Illustration of \ref{['lem:short-path-segments']}.
• Figure 4: Illustration of the construction in the proof of \ref{['lem-whard-induced']}.

Theorems & Definitions (65)

• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Claim 1
• proof