Geometric lower bound for the connectivity penalty

Derive a geometric lower bound for the connectivity penalty \(\rho\) above the \((\beta+1)\)-connectivity threshold, so that the approximation guarantee becomes a fully unconditional computable bound on non-degenerate instances.

Background

The approximation guarantee contains the factor ρ\rho, defined as the worst-case fraction of greedy marginal value retained when enforcing (β+1)(\beta+1)-vertex connectivity. The experiments measure ρ\rho across communication densities and suggest favorable behavior above the connectivity threshold, but the paper does not provide a geometric lower bound. Such a bound would make the penalty computable from geometric or communication parameters and remove this remaining dependence from the unconditional guarantee on non-degenerate instances.

References

Several directions remain open. First, the squared resilience factor is established under the intra-stage hypothesis of Proposition~\ref{prop:independence} and corroborated empirically. A proof that this hypothesis holds unconditionally for disk-graph geometries under bounded motion would bolster the idea. Second, deriving a geometric lower bound for $\rho$ %above the $(\beta{+}1)$-connectivity threshold would convert the penalty into a computable quantity, making eq:main-bound a fully unconditional guarantee on non-degenerate instances.

— Failure-Resilient Multi-Robot Coordination with $k$-Connectivity Constrained Submodular Maximization  (2610.13049 - Hwang et al., 9 Oct 2026) in Section Conclusion and Future Work