Abstract
The subset feedback vertex set problem generalizes the classical feedback vertex set problem and asks, for a given undirected graph G = (V, E), a set S ⊆ V, and an integer k, whether there exists a set X of at most k vertices such that no cycle in G ź X contains a vertex of S. It was independently shown by Cygan et al. (ICALP '11, SIDMA '13) and Kawarabayashi and Kobayashi (JCTB '12) that subset feedback vertex set is fixed-parameter tractable for parameter k. Cygan et al. asked whether the problem also admits a polynomial kernelization. We answer the question of Cygan et al. positively by giving a randomized polynomial kernelization for the equivalent version where S is a set of edges. In a first step we show that edge subset feedback vertex set has a randomized polynomial kernel parameterized by |S| + k with źź(|S|2k)$\mathcal {O}(|S|^{2}k)$ vertices. For this we use the matroid-based tools of Kratsch and Wahlstrom (FOCS '12) that for example were used to obtain a polynomial kernel for s-multiway cut. Next we present a preprocessing that reduces the given instance (G, S, k) to an equivalent instance (Gź, Sź, kź) where the size of Sź is bounded by źź(k4)$\mathcal {O}(k^{4})$. These two results lead to a randomized polynomial kernel for subset feedback vertex set with źź(k9)$\mathcal {O}(k^{9})$ vertices.
Highlights
IntroductionUsing the tree-like structure, a replacement argument can be found, implying that dominant solutions must create many components in (G − X) − S containing vertices of T and be well connected to them
In the subset feedback vertex set problem we are given an undirected graph G = (V, E), a set of vertices S ⊆ V, and an integer k, and have to determine whether there is a set X of at most k vertices that intersects all cycles that contain at least one vertex of S
Because we can choose S = V, this is a generalization of the well-studied feedback vertex set problem where, given G and k, we have to determine whether some set X of at most k vertices intersects all cycles in G. feedback vertex set has been extensively studied in parameterized complexity: It is known to be fixed-parameter tractable (FPT) with parameter k, i.e., solvable in time f (k) · |V |c, and after a series of improvements the fastest known algorithms take deterministic time O∗(3.619k) [11] and randomized time O∗(3k) [2]
Summary
Using the tree-like structure, a replacement argument can be found, implying that dominant solutions must create many components in (G − X) − S containing vertices of T and be well connected to them This allows to set up a gammoid on G − S with sources T and apply, as in [12], a result of Lovász [14] on representative sets in (linear) matroids that is guaranteed to generate a superset of X. A reduction of the number of S-edges is a crucial ingredient in the FPT algorithm for edge subset fvs by Cygan et al [3] They achieve |S| ∈ O(k3), but it is in a slightly more favorable setting: Using iterative compression, it suffices to solve the task of finding a solution X of size k when given a solution X of size k + 1. Proofs omitted in this extended abstract can be found in Hols and Kratsch [9]
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