Inapproximability of Minimum r-Cover

Determine lower bounds on the inapproximability of the minimum r-cover problem for the hard cases in which the radius r is a non-unit fraction.

Background

The paper contrasts the classical 2-approximation for minimum vertex cover with known hardness results and asks how these approximation barriers transfer to continuous minimum r-cover. The question is restricted to non-unit-fraction radii, for which the cited work establishes NP-hardness.

References

We suggest the following straightforward open question: For the hard cases of the problem (with the radius being a non-unit fraction, see~Hartmann et al), what are the lower bounds for the inapproximability of the minimum $r$-cover problem?

— Open Problems in Continuous Graphs  (2501.14554 - Grigoriev et al., 24 Jan 2025) in Section 3, subsection “Vertex cover,” paragraph “(In)approximability”