Inapproximability of Minimum Vertex Cover

Last time we examined a generic approach for inapproximability results based on the Unique Games Conjecture. Before, we had already shown that approximating MAX-3-LIN to within a constant factor larger than 12 is NP-hard. To do this we used a tweaked version of our dictatorship test that we came up with earlier in the semester. Last time we (re)proved that approximating MAX-3-LIN to within a constant larger than 1 2 is UG-hard. The latter is a weaker statement than the earlier NP-hardness result, but the argument used the dictatorship test as a blackbox. In this lecture we show that when we replace the dictatorship test by a noise sensitivity test, then we obtain that MAX-CUT is UG-hard to approximate to within any factor larger than ρGW , where ρGW refers to the approximation ratio achieved by the Goemans-Williamson algorithm. The numerical value of ρGW is approximately .878. We end the lecture wiht a proof sketch that approximating MIN-VC to within any constant factor smaller than 2 is UG-hard.