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Subset feedback vertex set is fixed-parameter tractable. (English) Zbl 1334.68096
Aceto, Luca (ed.) et al., Automata, languages and programming. 38th international colloquium, ICALP 2011, Zurich, Switzerland, July 4–8, 2011. Proceedings, Part I. Berlin: Springer (ISBN 978-3-642-22005-0/pbk). Lecture Notes in Computer Science 6755, 449-461 (2011).
Summary: The classical Feedback Vertex Set problem asks, for a given undirected graph $$G$$ and an integer $$k$$, to find a set of at most $$k$$ vertices that hits all the cycles in the graph $$G$$. Feedback Vertex Set has attracted a large amount of research in the parameterized setting, and subsequent kernelization and fixed-parameter algorithms have been a rich source of ideas in the field.
In this paper we consider a more general and difficult version of the problem, named Subset Feedback Vertex Set (Subset-FVS in short) where an instance comes additionally with a set $$S \subseteq V$$ of vertices, and we ask for a set of at most $$k$$ vertices that hits all simple cycles passing through $$S$$. Because of its applications in circuit testing and genetic linkage analysis Subset-FVS was studied from the approximation algorithms perspective by G. Even et al. [SIAM J. Discrete Math. 13, No. 2, 255–267 (2000; Zbl 0941.68057); SIAM J. Comput. 30, No. 4, 1231–1252 (2000; Zbl 0973.05073)].
The question whether the Subset-FVS problem is fixed-parameter tractable was posed independently by Kawarabayashi and Saurabh in 2009. We answer this question affirmatively. We begin by showing that this problem is fixed-parameter tractable when parametrized by $$|S|$$. Next we present an algorithm which reduces the given instance to $$2^{k } n ^{O(1)}$$ instances with the size of $$S$$ bounded by $$O(k ^{3})$$, using kernelization techniques such as the 2-Expansion Lemma, Menger’s theorem and Gallai’s theorem. These two facts allow us to give a $$2^{O(k \log k)} n ^{O(1)}$$ time algorithm solving the Subset Feedback Vertex Set problem, proving that it is indeed fixed-parameter tractable.
For the entire collection see [Zbl 1217.68003].

##### MSC:
 68Q25 Analysis of algorithms and problem complexity 05C85 Graph algorithms (graph-theoretic aspects) 68R10 Graph theory (including graph drawing) in computer science
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