Linear-query non-adaptive clustering or deterministic super-linear lower bound

Determine whether there exists a randomized non-adaptive algorithm that exactly recovers an arbitrary k-clustering of an n-point universe using subset queries in a linear O(n) number of queries; alternatively, establish a super-linear query lower bound for deterministic non-adaptive algorithms for exact k-clustering under the same subset-query model where a query on S ⊆ U returns the number of clusters intersecting S.

Background

The paper introduces non-adaptive clustering with subset queries and provides near-linear randomized algorithms, including an O(n * log k * (log k + log log n)^2) algorithm, an O(n * log log n * k * log k) algorithm for small k, and a 2-round algorithm with O(n * log k) queries. Despite these advances, it remains unknown whether strictly linear O(n) query complexity is achievable in the non-adaptive setting.

Prior work shows severe limitations for non-adaptive pairwise queries (Ω(n^2) even for k=3), motivating the use of larger subset queries. The authors explicitly note that they do not know whether an O(n) non-adaptive algorithm exists or whether a super-linear lower bound can be proven for deterministic non-adaptive algorithms.