Assuming that NP ⊈ ∩_ε > 0 BPTIME(2^n^ε), we show that GraphMin-Bisection, Densest Subgraph and Bipartite Clique have no PTAS.

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.