Most Classic Problems Remain NP-Hard on Relative Neighborhood Graphs and Their Relatives

Authors Pascal Kunz , Till Fluschnik , Rolf Niedermeier , Malte Renken

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Author Details

Pascal Kunz
  • Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
Till Fluschnik
  • Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
Rolf Niedermeier
  • Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
Malte Renken
  • Algorithmics and Computational Complexity, Technische Universität Berlin, Germany


This work is based on the first author’s master’s thesis. In memory of Rolf Niedermeier, our colleague, friend, and mentor, who sadly passed away before this paper was published.

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Pascal Kunz, Till Fluschnik, Rolf Niedermeier, and Malte Renken. Most Classic Problems Remain NP-Hard on Relative Neighborhood Graphs and Their Relatives. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 29:1-29:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Proximity graphs have been studied for several decades, motivated by applications in computational geometry, geography, data mining, and many other fields. However, the computational complexity of classic graph problems on proximity graphs mostly remained open. We study 3-Colorability, Dominating Set, Feedback Vertex Set, Hamiltonian Cycle, and Independent Set on the following classes of proximity graphs: relative neighborhood graphs, Gabriel graphs, and relatively closest graphs. We prove that all of the aforementioned problems remain NP-hard on these graphs, except for 3-Colorability and Hamiltonian Cycle on relatively closest graphs, where the former is trivial and the latter is left open. Moreover, for every NP-hard case we additionally show that no 2^{o(n^{1/4})}-time algorithm exists unless the Exponential-Time Hypothesis (ETH) fails, where n denotes the number of vertices.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Problems, reductions and completeness
  • Proximity Graphs
  • Relatively Closest Graphs
  • Gabriel Graphs
  • 3-Colorability
  • Dominating Set
  • Feedback Vertex Set
  • Hamiltonian Cycle
  • Independent Set
  • Exponential-Time Hypothesis


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