Close the competitive-ratio gap for adversarial MLA with delays

Determine the optimal competitive ratio for online Multi-Level Aggregation (MLA) with linear delays in the adversarial arrival model by closing the gap between the current O(d^2) upper bound (for tree depth d) and the current lower bound of 4, which holds even when the tree is a path with the root at an endpoint.

Background

The paper reviews that the best known competitive ratio for adversarial MLA with delays is O(d^2) due to Azar and Touitou (FOCS 2019), while the strongest lower bound is 4 from Bienkowski et al. (ESA 2016), even for the path case with the root at an endpoint. This substantial disparity motivates the need to refine either the upper or the lower bound.

The authors emphasize that this gap persists and pose it explicitly as an open question in the introduction, distinguishing their stochastic-results focus from the unresolved adversarial benchmark.