Improving the LPT approximation ratio for the Drone Warehouse Problem

Determine whether the approximation ratio of the Longest Processing Time First heuristic for the Drones Warehouse Problem can be improved to a constant strictly less than the golden ratio φ ≈ 1.62.

Background

The paper introduces the Drones Warehouse Problem (DWP), a scheduling generalization with heterogeneous drone speeds and battery constraints, and proves that an LPT-based algorithm achieves a φ-approximation with near-linear runtime. Since DWP generalizes uniform machine scheduling, known lower bounds (e.g., 1.54 for LPT on USP) carry over, but the current upper bound is φ.

The explicit question seeks to tighten this bound: either find an analysis or algorithm yielding an approximation ratio strictly below φ, or demonstrate that φ is optimal for LPT on DWP.