Parameterized Complexity of Feedback Vertex Sets on Hypergraphs
A feedback vertex set in a hypergraph H is a set of vertices S such that deleting S from H results in an acyclic hypergraph. Here, deleting a vertex means removing the vertex and all incident hyperedges, and a hypergraph is acyclic if its vertex-edge incidence graph is acyclic. We study the (parameterized complexity of) the Hypergraph Feedback Vertex Set (HFVS) problem: given as input a hypergraph H and an integer k, determine whether H has a feedback vertex set of size at most k. It is easy to see that this problem generalizes the classic Feedback Vertex Set (FVS) problem on graphs. Remarkably, despite the central role of FVS in parameterized algorithms and complexity, the parameterized complexity of a generalization of FVS to hypergraphs has not been studied previously. In this paper, we fill this void. Our main results are as follows
- HFVS is W[2]-hard (as opposed to FVS, which is fixed parameter tractable).
- If the input hypergraph is restricted to a linear hypergraph (no two hyperedges intersect in more than one vertex), HFVS admits a randomized algorithm with running time 2^{𝒪(k³log k)}n^{𝒪(1)}.
- If the input hypergraph is restricted to a d-hypergraph (hyperedges have cardinality at most d), then HFVS admits a deterministic algorithm with running time d^{𝒪(k)}n^{𝒪(1)}. The algorithm for linear hypergraphs combines ideas from the randomized algorithm for FVS by Becker et al. [J. Artif. Intell. Res., 2000] with the branching algorithm for Point Line Cover by Langerman and Morin [Discrete & Computational Geometry, 2005].
feedback vertex sets
hypergraphs
FPT
randomized algorithms
Theory of computation~Parameterized complexity and exact algorithms
18:1-18:15
Regular Paper
We thank the anonymous referees of an earlier version of the paper. Their comments helped us a lot in improving the paper.
Pratibha
Choudhary
Pratibha Choudhary
Indian Institute of Technology Jodhpur, Jodhpur, India
Lawqueen
Kanesh
Lawqueen Kanesh
Institute of Mathematical Sciences, HBNI, Chennai, India
Daniel
Lokshtanov
Daniel Lokshtanov
University of California Santa Barbara, Santa Barbara, USA
Fahad
Panolan
Fahad Panolan
Indian Institute of Technology Hyderabad, India
Saket
Saurabh
Saket Saurabh
Institute of Mathematical Sciences, HBNI, Chennai, India
University of Bergen, Norway
Received funding from European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no. 819416), and Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18.
10.4230/LIPIcs.FSTTCS.2020.18
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Pratibha Choudhary, Lawqueen Kanesh, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh
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