Hardness of approximation for crossing number

TL;DR

This paper proves that, assuming P≠NP, it is computationally hard to approximate the crossing number within any constant factor greater than 1, even for 3-regular graphs.

Contribution

It establishes the hardness of approximating the crossing number within any constant factor for 3-regular graphs under standard complexity assumptions.

Findings

No c-approximation algorithm exists for the crossing number if P≠NP.

Hardness result applies even to 3-regular graphs.

Supports the computational difficulty of crossing number approximation.

Abstract

We show that, if P\not=NP, there is a constant c > 1 such that there is no c-approximation algorithm for the crossing number, even when restricted to 3-regular graphs.