Ruling out PTAS for graph min-bisection, densest subgraph and bipartite clique

Assuming that NP not subset of or equal to boolean AND(epsilon)>0 BPTIME(2(nepsilon)), we show that Graph Min-Bisection, Densest Subgraph and Bipartite Clique have no PTA,S. We give a reduction from the Minimum Distance Of Code Problem (MDC). Starting with an instance of MDC, we build a Quasi-random PCP that suffices to prove the desired inapproximability results. In a Quasi-random PCP the query pattern of the verifier looks random in some precise sense. Among the several new techniques introduced, we give a way of certifying that a given polynomial belongs to a given subspace of polynomials. As is important for our purpose, the certificate itself happens to be another polynomial and it can be checked by reading a constant number of its values.