Metric Dimension Parameterized by Feedback Vertex Set and Other Structural Parameters
For a graph G, a subset S ⊆ V(G) is called a resolving set if for any two vertices u,v ∈ V(G), there exists a vertex w ∈ S such that d(w,u) ≠ d(w,v). The Metric Dimension problem takes as input a graph G and a positive integer k, and asks whether there exists a resolving set of size at most k. This problem was introduced in the 1970s and is known to be NP-hard [GT 61 in Garey and Johnson’s book]. In the realm of parameterized complexity, Hartung and Nichterlein [CCC 2013] proved that the problem is W[2]-hard when parameterized by the natural parameter k. They also observed that it is FPT when parameterized by the vertex cover number and asked about its complexity under smaller parameters, in particular the feedback vertex set number. We answer this question by proving that Metric Dimension is W[1]-hard when parameterized by the feedback vertex set number. This also improves the result of Bonnet and Purohit [IPEC 2019] which states that the problem is W[1]-hard parameterized by the treewidth. Regarding the parameterization by the vertex cover number, we prove that Metric Dimension does not admit a polynomial kernel under this parameterization unless NP ⊆ coNP/poly. We observe that a similar result holds when the parameter is the distance to clique. On the positive side, we show that Metric Dimension is FPT when parameterized by either the distance to cluster or the distance to co-cluster, both of which are smaller parameters than the vertex cover number.
Metric Dimension
Parameterized Complexity
Feedback Vertex Set
Theory of computation~Parameterized complexity and exact algorithms
51:1-51:15
Regular Paper
Research supported by the European Research Council (ERC) consolidator grant No. 725978 SYSTEMATICGRAPH.
https://arxiv.org/abs/2206.15424
The authors would like to thank Florent Foucaud for pointing us to Gutin et al. [G. Z. Gutin et al., 2020]. The article contains a result that subsumes our result conditionally refuting the polynomial kernel for Metric Dimension parameterized by the vertex cover number.
Esther
Galby
Esther Galby
CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Liana
Khazaliya
Liana Khazaliya
Saint Petersburg State University, Saint Petersburg, Russia
Fionn
Mc Inerney
Fionn Mc Inerney
CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Roohani
Sharma
Roohani Sharma
Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Prafullkumar
Tale
Prafullkumar Tale
CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
10.4230/LIPIcs.MFCS.2022.51
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Esther Galby, Liana Khazaliya, Fionn Mc Inerney, Roohani Sharma, and Prafullkumar Tale
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