Competition parameter for light vertex fault-tolerant spanners

Determine the smallest function $\mathcal{C}(f)$ such that every n-node weighted graph has an f-vertex fault-tolerant k-spanner with vertex-competitive lightness at most $O(\operatorname{poly}(f)\cdot\lambda(n,k+1))$, and in particular determine whether the edge-fault threshold $\mathcal{C}(f)=2f$ has a matching analogue for vertex faults.

Background

The paper proves that the minimum competition parameter is exactly 2f in the edge-fault setting and observes that the lower-bound argument gives a lower bound of 2f for vertex faults. It does not establish a matching upper bound for vertex fault-tolerant spanners. The proposed question asks for the optimal competition parameter under an analogous vertex-competitive lightness definition.

References

In addition to the obvious open question of determineg the precise bound on the achievable $2f$-competitive lightness, an interesting open problem left by this paper is to determine the appropriate competition parameter for light vertex fault tolerant (VFT) spanners.

— Light Edge Fault Tolerant Graph Spanners  (2502.10890 - Bodwin et al., 15 Feb 2025) in Section 5, “Future Directions: Light Vertex Fault Tolerant Spanners”