Improve the dependence on k in bounded-query-size algorithms

Improve the dependence on the number of clusters k in the query complexity of the non-adaptive k-clustering algorithm with bounded query size s ≤ √n, which currently uses \widetilde{O}(n^2 k/s^2) subset queries, aiming to reduce or eliminate the linear k factor while maintaining near-optimal dependence on s.

Background

For bounded query size s, the paper presents a non-adaptive algorithm with query complexity \widetilde{O}(n^2 k/s^2) (optimal up to logarithmic factors for constant k) and another with \widetilde{O}(n^2/s), each trading off dependence on k and s. The lower bound is Ω(max(n^2/s^2, n)) even for k=3.

The authors explicitly identify improving the dependence on k in their \widetilde{O}(n^2 k/s^2) algorithm as an open problem, highlighting a gap between current upper bounds and the lower bound when k is not constant.