The value of strong inapproximability results for clique

We consider approximations of the form n ~-°(1) for the Maximum Clique problem, where n is the number of vertices in the input graph and where the "o(1)" term goes to zero as n increases. We show that sufficiently strong negative results for such problems, which we call strong inapproximability results, have interesting consequences for exact computation. In particular, we show that for some such clique approximation problems that seem likely to require superpolynomial time in view of the results of Engebretsen and Holmerin (Manuscript, 1999), even certain low-degree polynomial lower bounds on their complexity will prove that N P ~ P. Our approach also leads to approximation algorithms: e.g., for (weighted) Maximum Clique and Maximum Independent Set, and for a class of maximization problems that includes the packing integer programs. A simple sampling method underlies most of our results.