Sublogarithmic awake complexity for logarithmic-approximation MDS

Determine whether an $O(\log \Delta)$-approximate Minimum Dominating Set can be computed with $o(\log \Delta)$ awake complexity in the sleeping distributed model.

Background

The paper develops sleeping-model algorithms for Minimum Dominating Set (MDS) that achieve an $O(\log \Delta)$ approximation with $O(\log \Delta\,\log\log n)$ awake complexity, and in certain parameter regimes nearly $O(\log \Delta)$ awake complexity. The authors ask whether the awake complexity can be reduced below $\log \Delta$ without worsening the approximation factor.

The authors explain that a natural strategy in which nodes use their residual degree to skip stages fails in the worst case: the appendix gives a graph where many nodes experience wasted wake-ups in almost every stage. Consequently, they state that solving this problem appears to require fundamentally new ideas.