Constant-Factor Approximation for Maximum r-Independent Set

Determine whether, for every rational radius r>0, a polynomial-time approximation algorithm exists for the maximum r-independent set problem with a performance guarantee bounded by a constant.

Background

The paper notes that the continuous problem has polynomial-time solutions for some radii and gives a constant performance guarantee for rational radii at most 1/2. It asks whether a constant-factor approximation can be obtained uniformly for every rational positive radius, including radii for which exact computation may be hard.

References

The specific open question we ask in this paragraph reads: For any rational number $r>0$, is there a polynomial time approximation algorithm for the maximum $r$-independent set problem with performance guarantee bounded by a constant?

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