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

by   Pascal Kunz, et al.

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.


page 1

page 2

page 3

page 4


Feedback Vertex Set on Hamiltonian Graphs

We study the computational complexity of Feedback Vertex Set on subclass...

1-Extendability of independent sets

In the 70s, Berge introduced 1-extendable graphs (also called B-graphs),...

Computational Complexity of Three Central Problems in Itemset Mining

Itemset mining is one of the most studied tasks in knowledge discovery. ...

3-Coloring on Regular, Planar, and Ordered Hamiltonian Graphs

We prove that 3-Coloring remains NP-hard on 4- and 5-regular planar Hami...

On Computing the Hamiltonian Index of Graphs

The r-th iterated line graph L^r(G) of a graph G is defined by: (i) L^0(...

Sparse dominating sets and balanced neighborhood partitioning

Recent work in metagenomics constructs a partition of the assembly graph...

Complexity continuum within Ising formulation of NP problems

A promising approach to achieve computational supremacy over the classic...