Complexity for Algebraic Cover Radii

Determine the complexity of the minimum r-cover problem on continuous graphs for algebraic real radii r, particularly whether it is NP-hard for every algebraic radius that is not a unit fraction.

Background

The cited results establish that the minimum r-cover problem is in NP for rational radii, polynomially solvable for unit fractions, and NP-hard for rational non-unit fractions. The paper reports an open problem concerning algebraic real radii and states the cited authors’ conjecture that non-unit-fraction algebraic radii retain NP-hardness.

References

As an interesting open problem, Hartmann et al. suggest to study the complexity of the problem with algebraic real $r$. Such problems are also contained in $NP$, and the conjecture of Hartmann et al. is that the problem is $NP$-hard for all algebraic values $r$ that are not unit fractions.

— Open Problems in Continuous Graphs  (2501.14554 - Grigoriev et al., 24 Jan 2025) in Section 3, subsection “Vertex cover,” paragraph “Discretization, tractability and $NP$-certificates”