Most Classic Problems Remain NP-hard on Relative Neighborhood Graphs and their Relatives

TL;DR

This paper proves that many classic NP-hard problems remain computationally difficult on various proximity graph classes, with some exceptions, and establishes strong lower bounds based on ETH.

Contribution

It demonstrates NP-hardness of key problems on proximity graphs and provides exponential time lower bounds, advancing understanding of their computational complexity.

Findings

NP-hardness of 3-Colorability, Dominating Set, Feedback Vertex Set, and Independent Set on these graphs

3-Colorability is trivial on relatively closest graphs

No subexponential algorithms exist under ETH for these problems

Abstract

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 now study 3-Colorability, Dominating Set, Feedback Vertex Set, Hamiltonian Cycle, and Independent Set on the proximity graph classes relative neighborhood graphs, Gabriel graphs, and relatively closest graphs. We prove that all of the 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 ETH fails, where n denotes the number of vertices.