Ruling Out PTAS for Graph Min‐Bisection, Dense k‐Subgraph, and Bipartite Clique

Assuming that NP ⊆ ∩ε > 0 BPTIME(2n^ε), we show that graph min‐bisection, dense k‐subgraph, and bipartite clique have no polynomial time approximation scheme (PTAS). We give a reduction from the minimum distance of code (MDC) problem. Starting with an instance of MDC, we build a quasi‐random probabilistically checkable proof (PCP) that suffices to prove the desired inapproximability results. In a quasi‐random PCP, the query pattern of the verifier looks random in a certain precise sense. Among the several new techniques we introduce, the most interesting one gives a way of certifying that a given polynomial belongs to a given linear subspace of polynomials. As is important for our purpose, the certificate itself happens to be another polynomial, and it can be checked probabilistically by reading a constant number of its values.