Characterize the optimal existential satisfaction ratio for OPI

Characterize the largest satisfaction ratio s^* guaranteed to exist for every Optimal Polynomial Intersection instance of rate R with input-list density at least ρ, including the parameter regimes in which the current lower bound V_ρ(R) may be improved.

Background

Optimal Polynomial Intersection (OPI) asks for a low-degree polynomial satisfying as many coordinate-wise list constraints as possible. The paper determines perfect-solution regimes and proves a lower bound V_ρ(R) in other regimes, but does not establish the exact worst-case satisfaction ratio throughout the full parameter space. The authors specifically suggest that the lower bound may be improvable by a positive, possibly arbitrarily small, amount in regimes where their results do not yield perfect solutions.

References

What is the ``correct'' satisfaction ratio for an OPI instance of rate $R$? That is, given $R \in (0,1)$ and $\rho \in (0,1)$, what is the largest $s*$ so that, for any OPI instance of rate $R$ and input lists with density $\rho$, there exists a solution with satisfaction ratio at least $s*$? In some parameter regimes, we have shown that $s* = 1$, but in the other parameter regimes, we have only given bounds on $s*$. In particular, we have shown that $s* \geq V_\rho(R)$, but our results \cref{thm:main-opi}(2) suggest that one can do better, at least by some arbitrarily small $\delta > 0$.

— Improved Local Leakage Resilience of Shamir Secret Sharing and Worst-Case Optimal Polynomial Intersection  (2610.12357 - Sun et al., 8 Oct 2026) in Section ‘Open Problems’