Conjectured Ω(k polylog k) randomized lower bound for Robust Max Selection

Prove an Ω(k polylog k) lower bound on the number of pairwise comparison queries required by any randomized algorithm in the Robust Max Selection model—where n elements include k adversarially corrupted elements, comparisons are obtained via a black-box oracle that may answer arbitrarily on corrupted elements, and the algorithm must output a set of exactly 2k + 1 elements that contains the uncorrupted maximum—with at least constant success probability.

Background

The paper introduces a new comparison-based model for algorithm design under adversarially corrupted inputs. Among n elements, k are corrupted and may respond arbitrarily to pairwise comparison queries, potentially creating cycles. Algorithms must output a set that contains the true (uncorrupted) maximum; the minimal feasible output size is min{n, 2k + 1}, and the work focuses on the case n > 2k + 1 with output size exactly 2k + 1.

For randomized algorithms, the authors prove an Ω(n) lower bound (e.g., at most n/4 queries cannot achieve success probability > 3/4 under a specific hard instance) and present a two-stage algorithm using O(n + k polylog k) queries with high probability. To further tighten the bounds, they conjecture an additional Ω(k polylog k) lower bound, motivated by a counting argument on the hard instance (which yields an Ω(k log n) deterministic lower bound) and their belief that a similar reasoning extends to randomized settings.