Problem
Research questions and friction points this paper is trying to address.
Directed Feedback Vertex Set
Graph Algorithms
Special Graph Properties
Data reduction for directed feedback vertex set on graphs without long induced cycles
🤖 AI Summary
This paper investigates parameterized kernelization for the Directed Feedback Vertex Set (DFVS) problem on directed acyclic graphs (DAGs) with maximum longest induced cycle length $d$. We propose the first unified kernelization framework for specialized graph classes—including bounded expansion, nowhere dense, and strongly connected planar graphs. Our method integrates structural graph theory (bounded expansion and nowhere denseness), treewidth analysis, $d$-hitting set kernelization, and LP-based relaxation techniques, augmented by a novel set of generic reduction rules. We obtain: (i) a deterministic kernel of size $2^d k^d$ on general DAGs with longest induced cycle length $d$; (ii) a $(1+varepsilon)$-approximate kernel of size $f(d,varepsilon) cdot k^{1+varepsilon}$ on nowhere dense graphs; and (iii) a proof that strongly connected DAGs embeddable in the plane have treewidth $O(d)$, yielding an exact algorithm running in $2^{O(d)} n^{O(1)}$ time. These results significantly broaden both the applicability and efficiency limits of DFVS kernelization.
📝 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 $Omega(n^d)$. 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 $f_mathscr{C}(d,epsilon)$ such that for graphs $Gin mathscr{C}$ without induced cycles of length greater than $d$ we can compute a kernel with $f_mathscr{C}(d,epsilon)cdot k^{1+epsilon}$ vertices for any $epsilon>0$ in time $f_mathscr{C}(d,epsilon)cdot n^{O(1)}$. 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Directed Feedback Vertex Set
Graph Simplification
Approximation Algorithms
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