Polynomial Certificates for Maximum r-Independent Sets

Establish whether, for every rational radius r>0, the maximum r-independent set problem on general continuous graphs has a polynomial-size certificate.

Background

For small rational radii, the cited prior work provides certificates checked using a linear program. The paper explains that this approach fails for larger radii because it assumes that every edge contains a point of the r-independent set. The unresolved issue is whether another polynomial-size certificate exists for arbitrary rational radii on general continuous graphs.

References

Thus, a specific open question is: Given any rational number $r>0$, is there a polynomial size certificate for the maximum $r$-independent set problem on general continuous graphs?

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