Randomized r-round lower bounds for pairwise partition learning

Establish, for every integer r ≥ 1, a lower bound on the number of pairwise same-set queries required by any randomized r-round algorithm that exactly learns an unknown partition of n elements into at most k sets. Specifically, prove that any randomized r-round algorithm must use at least Ω((1/r)·n^{1+1/(2^r−1)}·k^{1−1/(2^r−1)}) pairwise same-set queries, analogous to the deterministic r-round lower bound proved for this problem.

Background

The paper studies the problem of learning an unknown k-partition of n elements using pairwise same-set queries under bounded rounds of adaptivity r. It provides tight deterministic upper and lower bounds that interpolate smoothly between non-adaptive (r=1, Θ(n^2)) and fully adaptive (Θ(nk)) regimes, with the deterministic r-round lower bound stated as Ω((1/r)·n^{1+1/(2^r−1)}·k^{1−1/(2^r−1)}).

While the deterministic lower bound is established using a graph-based approach and Turán’s theorem, the authors note that extending this lower bound to arbitrary randomized algorithms remains unresolved and likely requires new technical ideas. This open problem asks for analogous r-round lower bounds in the randomized setting.