Parameterized Complexity of Geodetic Set
A vertex set S of a graph G is geodetic if every vertex of G lies on a shortest path between two vertices in S. Given a graph G and k ∈ ℕ, the NP-hard Geodetic Set problem asks whether there is a geodetic set of size at most k. Complementing various works on Geodetic Set restricted to special graph classes, we initiate a parameterized complexity study of Geodetic Set and show, on the negative side, that Geodetic Set is W[1]-hard when parameterized by feedback vertex number, path-width, and solution size, combined. On the positive side, we develop fixed-parameter algorithms with respect to the feedback edge number, the tree-depth, and the modular-width of the input graph.
NP-hard graph problems
Shortest paths
Tree-likeness
Parameter hierarchy
Data reduction
Integer linear programming
Theory of computation~Parameterized complexity and exact algorithms
20:1-20:14
Regular Paper
A full version of the paper is available at https://arxiv.org/abs/2001.03098.
We thank Lucia Draque Penso (Ulm University) for suggesting studying Geodetic Set from a view of parameterized complexity, and we thank André Nichterlein and Rolf Niedermeier (both TU Berlin) for helpful feedback and discussion. We are also grateful to an anonymous reviewer for suggesting that the ILP instances in Section 4 can be solved more efficiently.
Leon
Kellerhals
Leon Kellerhals
Technische Universität Berlin, Faculty IV, Algorithmics and Computational Complexity, Germany
https://kellerhals.io
https://orcid.org/0000-0001-6565-3983
Tomohiro
Koana
Tomohiro Koana
Technische Universität Berlin, Faculty IV, Algorithmics and Computational Complexity, Germany
https://tomohirokoana.github.io/
https://orcid.org/0000-0002-8684-0611
Partially supported by the DFG projects FPTinP (NI 369/16) and MATE (NI 369/17).
10.4230/LIPIcs.IPEC.2020.20
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Leon Kellerhals and Tomohiro Koana
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