Tight Stretch Bounds for Fixed-Size Tree Covers in General Metrics

Determine a nearly tight stretch bound, conjecturally of order \(\Omega(n^{1/k})\), for tree covers of general metrics using a fixed constant number \(k\geq 3\) of trees.

Background

The paper discusses the fixed cost paradigm, in which the number of trees is fixed and the goal is to minimize the resulting stretch. For general metrics, existing upper and lower bounds are close for two trees but leave a substantial gap for every constant $k\geq 3$. The authors indicate that the expected lower-bound order is likely $\Omega(n^{1/k})$, but a nearly tight bound remains unresolved.