Extending new lower bounds to stronger log^β Δ-approximation regimes

Extend the non-signaling (and derived quantum-LOCAL) locality lower bounds for minimum dominating set proved in this work to the regime where the approximation factor is (log Δ)^β for arbitrary β>0, matching the Kuhn–Moscibroda–Wattenhofer lower bound’s dependence on β for such approximations.

Background

The paper proves lower bounds for O(log Δ) and for O((log Δ)β) for some β∈(0,1). By contrast, the classic Kuhn–Moscibroda–Wattenhofer bound applies to (log Δ)β for any β>0, albeit with different trade-offs. The authors therefore point out the gap between their current results and this stronger regime and explicitly leave it open whether their techniques can be pushed to all β>0.

Resolving this would unify the new non-signaling (and quantum-LOCAL) lower bounds with the parameter range of the established LOCAL-model results for degree-dependent approximations.

References

We note that the KMW bound in fact holds for even stronger approximation ratios beyond $\log \Delta$, in particular for any $\beta>0$ one gets a lower bound of $\Omega \left(\min\left{\frac{\log \Delta}{\beta \log\log \Delta}, \sqrt{\frac{\log n}{\beta \log \log n}\right}\right)$ against $\log{\beta} \Delta$-approximation. We leave as an open problem whether our results can be extended to this regime.

— Non-Signaling Locality Lower Bounds for Dominating Set  (2604.02582 - Fleming et al., 2 Apr 2026) in Section 1: Introduction (after Table 1)