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

• Authors: Subhash Khot
• Venue: SIAM Journal on Computing
• Year: 2006

Assuming that NP $\not\subseteq$ $\cap_{\epsilon 0}$ BPTIME($2^{n^\epsilon}$), 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.