A Randomized Polynomial Kernel for Subset Feedback Vertex Set

Authors Eva-Maria C. Hols, Stefan Kratsch

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Eva-Maria C. Hols
Stefan Kratsch

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Eva-Maria C. Hols and Stefan Kratsch. A Randomized Polynomial Kernel for Subset Feedback Vertex Set. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 43:1-43:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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 subseteq V, and an integer k, whether there exists a set X of at most k vertices such that no cycle in G-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 O(|S|^2k) vertices. For this we use the matroid-based tools of Kratsch and Wahlstrom (FOCS'12). Next we present a preprocessing that reduces the given instance (G,S,k) to an equivalent instance (G',S',k') where the size of S' is bounded by O(k^4). These two results lead to a polynomial kernel for SUBSET FEEDBACK VERTEX SET with O(k^9) vertices.
  • parameterized complexity
  • kernelization
  • subset feedback vertex set


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