Constant-factor guarantees under constant slackness and extension

Determine whether there exists an online algorithm for the online block packing problem that achieves a constant-factor approximation to the offline optimal social welfare when both the average-capacity slackness Δ and the extension Γ are fixed constants independent of the resource dimension m; alternatively, ascertain whether a constant-factor approximation exists that only inflates the per-block maximum capacity by a constant factor while leaving the long-run average capacity unchanged.

Background

The paper’s general-case algorithm achieves approximation with slackness and extension parameters that scale as O(log m). It is unknown whether similar constant-factor guarantees are achievable when both augmentation parameters are absolute constants independent of m, which would be more attractive in blockchain applications.

An easier target proposed by the authors is to obtain a constant-factor approximation by allowing only a constant inflation of the per-block maximum capacity without enforcing average-capacity control, which could simplify practical deployment but may be computationally challenging.