2026-08-10T22:31:17Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:5744 2024-03-06T10:36:58Z ddc:004 open_access

A Randomized Polynomial Kernel for Subset Feedback Vertex Set Hols, Eva-Maria C. Kratsch, Stefan parameterized complexity kernelization subset feedback vertex set 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Eva-Maria C. Hols and Stefan Kratsch 2016 Is Part Of LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2016.43 urn:nbn:de:0030-drops-57448 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.43 eng https://creativecommons.org/licenses/by/3.0/legalcode