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

Assuming that NP ⊄ ∩ _ε>0 BPTIME(2 ^nε), we show that Graph Min-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.