Close the non-interactive STV exponential-dependence gap

Determine whether the gap between the non-interactive local-differential-privacy upper-bound dependence $2^{O(m)}$ and lower-bound dependence $2^{\Omega(\sqrt{m})}$ for the decisive margin of Single Transferable Vote can be closed.

Background

For non-interactive local differential privacy, the paper gives an upper bound for Single Transferable Vote whose dependence on the number of candidates is exponential of order 2O(m)2^{O(m)}, arising because every voter must privatize plurality statistics for all possible active candidate sets. The corresponding lower bound has dependence only 2Ω(m)2^{\Omega(\sqrt{m})}, obtained through reductions from high-dimensional marginal release. The authors explicitly identify closing this asymptotic gap as unresolved.

References

On the other hand, the more interesting remaining gap here is the $2{O(m)}$ term versus the $2{\Omega(\sqrt{m})}$ term in our upper and lower bounds, respectively. Closing this gap is an interesting open question.

— Decisive Margins in Differentially Private Voting  (2608.18772 - Hillebrand et al., 19 Aug 2026) in Section 3, subsection “STV,” immediately following Corollary 3.10