Extending Ω(log n) locality lower bounds to O(log Δ)-approximation in LOCAL

Determine whether the Ω(log n) locality lower bound for minimum dominating set proved by Chang and Li for constant-factor approximation in the LOCAL model extends to O(log Δ)-approximation algorithms on n-vertex graphs with maximum degree Δ, thereby matching the performance of known LOCAL algorithms that achieve (1+ε)·log Δ approximation.

Background

The paper reviews two strands of prior results for minimum dominating set in distributed computing: an Ω(log n) lower bound in the LOCAL model for some constant-factor approximation (Chang–Li), and algorithms/lower bounds that depend on the maximum degree Δ (Kuhn–Moscibroda–Wattenhofer). The authors note a gap: for small Δ, known degree-dependent lower bounds are weak and do not reach Ω(log n). This motivates asking whether the stronger Ω(log n) lower bound can also be proved for the more permissive O(log Δ)-approximation ratios that match the best known algorithms.

The authors make substantial progress toward this direction in the non-signaling model, but the exact question of achieving the full Ω(log n) for O(log Δ)-approximation in LOCAL remains explicitly open as posed in the introduction.