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

TL;DR

This paper develops kernelization techniques for the Directed Feedback Vertex Set problem on graphs without long cycles, providing efficient algorithms for special graph classes and new reduction rules.

Contribution

It introduces new kernelization bounds for DFVS on graphs without long cycles, especially in nowhere dense classes and planar graphs, and proposes a new data reduction rule.

Findings

Kernel with at most 2^dk^d vertices for bounded cycle length graphs.

Polynomial kernels for graphs in nowhere dense classes.

DFVS on planar graphs without long cycles solvable in fixed-parameter time.

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 $G\in \mathscr{C}$ 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.