Complexity for Nonreduced Rational Radii

Determine the computational complexity of the maximum r-independent set problem on continuous graphs when r=a/(2b), with integers a and b having a common divisor greater than one.

Background

The paper studies maximum r-independent sets in continuous graphs, where selected points must be pairwise at distance at least 2r. Prior results cited in the paper establish polynomial-time solvability for unit-fraction radii and NP-hardness for reduced fractions of the form a/(2b) with a≥3. The authors explicitly identify the case in which a and b are not coprime as unresolved.

References

If $a$ and $b$ have a common divisor larger than $1$, the complexity of the problem remains open.

— Open Problems in Continuous Graphs  (2501.14554 - Grigoriev et al., 24 Jan 2025) in Section 3, subsection “Maximum independent set,” paragraph “Discretization and tractability”