Data reduction for directed feedback vertex set on graphs without long induced cycles
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We study reduction rules for Directed Feedback Vertex Set (DFVS) on instances 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^3dk^d induced cycles of length at most d (which however, cannot be enumerated efficiently), 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; these are very general classes of sparse graphs, containing e.g. classes excluding a minor or a topological minor. We prove that for such classes without induced cycles of length greater than d we can compute a kernel with O_d(k) and O_d,ϵ(k^1+ϵ) vertices for any ϵ>0, respectively, in time O_d(n^O(1)) and O_d,ϵ(n^O(1)), respectively. The most restricted classes we consider are strongly connected planar graphs without any (induced or non-induced) long cycles. We show that these have bounded treewidth and hence DFVS on planar graphs without cycles of length greater than d can be solved in time 2^O(d)· n^O(1). We finally present a new data reduction rule for general DFVS and prove that the rule together with a few standard rules subsumes all the rules applied by 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.
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