Reduction of metric-sensitive logarithmic factors

Determine whether the $\log^2 n$ factors in the metric-sensitive competitive guarantee for Robustified Greedy can be reduced.

Background

Robustified Greedy achieves a metric-sensitive competitive ratio of $O(k^{1-1/d}\log^2 n)$ in fixed-dimensional Euclidean spaces, and more generally an input-sensitive guarantee involving a $\log^2 n$ factor. The paper establishes this bound but does not determine whether the logarithmic dependence is necessary or can be improved.

The authors identify reduction of these logarithmic factors as a direction for future work, making the unresolved issue explicit.