Optimal dependence on query size s for all k

Establish whether there exists a non-adaptive algorithm that exactly recovers an arbitrary k-clustering using subset queries of maximum size s with query complexity \widetilde{O}(n^2/s^2) for all k, thereby matching the Ω(max(n^2/s^2, n)) lower bound in general.

Background

When the query size is bounded by s, the paper proves a lower bound of Ω(max(n^2/s^2, n)) and gives algorithms achieving \widetilde{O}(n^2 k/s^2) for s ≤ √n and \widetilde{O}(n^2/s) for s ≤ n. For constant k, the first algorithm is nearly optimal up to logarithmic factors, but the dependence on k remains non-optimal in general.

The authors pose the open question of achieving the optimal s-dependence \widetilde{O}(n^2/s^2) uniformly for all k, which would close the gap between their upper bounds and the lower bound across all regimes of k.