Close the visible competitive-ratio gaps

Close the gaps between the deterministic visible-service lower bound of 3 and the upper bounds of 10 on finite lines, 12 on weighted trees, and 20 on arbitrary finite metrics, and determine whether randomization improves these competitive ratios.

Background

The paper establishes a universal deterministic lower bound of 3 for visible elective and automatic service, while its algorithms achieve ratios 10 on finite lines, 12 on weighted trees, and 20 on arbitrary finite metrics. The authors explicitly identify closing these quantitative gaps, including understanding the effect of randomization, as the first goal on the competitive axis.

References

Three coherent axes remain open. On the competitive axis, the first goal is to close the visible gaps between the lower bound $3$ and the upper bounds $10,12,20$, and to determine whether randomization improves them.

— Online Service with Per-Batch Maximum Delay  (2608.18577 - Lu et al., 19 Aug 2026) in Section 6, Conclusion and Future Directions

Online, the spatial-surrogate transfer principle asks whether efficiently computable structures better than the factor-two terminal MST can improve the polynomial ratio $20$.

— Online Service with Per-Batch Maximum Delay  (2608.18577 - Lu et al., 19 Aug 2026) in Section 6, Conclusion and Future Directions