Close the augmentation gap for price-based algorithms (log vs. double-log in value range)

Determine whether there exists a price-based online scheduling algorithm for transactions with per‑unit values in the range [1, H] that guarantees a constant fraction of the optimal social welfare under an average block size limit B using augmentation whose total amount Δ+Γ grows only as O(log log H), for example by constructing a useful variant of the EIP-1559 price update that achieves such double‑logarithmic growth in Δ+Γ.

Background

The paper proves that any price-based algorithm that achieves a constant fraction of the optimal welfare must have augmentation (the sum of slackness Δ and extension Γ) that grows at least as Ω(log log H) in the value range H. However, their positive results for EIP-1559 require augmentation that is O(log H), leaving an exponential gap between the lower and upper bounds.

The authors explicitly note that closing this gap is open and highlight the specific possibility of designing a variant of EIP-1559 whose augmentation grows only double‑logarithmically in H, while still obtaining a constant-factor welfare guarantee against the clairvoyant optimum with worst-case block size B.