Inapproximability of dominating set on power law graphs

We prove the first logarithmic lower bounds for the approximability of the Minimum Dominating Set problem for the case of connected (α,β)(α,β)-power law graphs for α being a size parameter and β   the power law exponent. We give also a best up to now upper approximation bound for this problem in the case of the parameters β>2β>2. We develop also a new functional method for proving lower approximation bounds and display a sharp approximation phase transition area between approximability and inapproximability of the underlying problems. Our results depend on a method which could be also of independent interest.