Polynomial-time computation of \(\tilde{\nu}^{(r)}(G)\)

Determine whether \(\tilde{\nu}^{(r)}(G)\) can be computed to fixed precision in polynomial time for every fixed level \(r\in\mathbb{N}\).

Background

For fixed rr, evaluating the proposed bound ν~(r)(G)\tilde{\nu}^{(r)}(G) can be reduced to a logarithmic number of SDP-feasibility problems through binary search. However, the paper notes that the complexity of general SDP feasibility is unresolved, and therefore does not establish a polynomial-time algorithm for this graph parameter.

References

Therefore, we pose an open question: can $\tilde{\nu}{(r)}(G)$ be computed (up to a fixed precision) in polynomial time for fixed $r\in \mathbb{N}$?.

— Low degree sum-of-squares bounds for the stability number: a copositive approach  (2509.04949 - Vargas et al., 5 Sep 2025) in Section 10, Conclusion and Discussion