Unconditional validity of the intra-stage hypothesis

Prove unconditionally that the intra-stage hypothesis of Proposition 1 holds for disk-graph geometries under bounded motion, thereby establishing the squared resilience factor without requiring that hypothesis as an additional condition.

Background

The paper’s squared approximation guarantee depends on the intra-stage hypothesis introduced in Proposition 1, which assumes that feasible S2 completions depend on the S1 bait selection only through the fixed endpoint set and that the connectivity penalty is conditionally independent of the specific S1 trajectories. The authors establish the squared factor under this hypothesis and report empirical support, but do not prove that the hypothesis necessarily holds for disk-graph communication geometries with bounded robot motion.

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.

— 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