Local Approximability of Minimum Dominating Set on Planar Graphs
  We show that there is no deterministic local algorithm (constant-time distributed graph algorithm) that finds a $(7-\epsilon)$-approximation of a minimum dominating set on planar graphs, for any positive constant $\epsilon$. In prior work, the best lower bound on the approximation ratio has been $5-\epsilon$; there is also an upper bound of $52$. 
Miikka Hilke, Christoph Lenzen, Jukka Suomela
cs.DC cs.DS