Free bits, PCPs and non-approximability-towards tight results

The first part of this paper presents new proof systems and improved non-approximability results. In particular we present a proof system for NP using logarithmic randomness and two amortized free bits, so that Max clique is hard within N^1/3 and chromatic number within N^1/5. We also show hardness of 38/37 for Max-3-SAT, 27/26 for vertex cover, 82/81 for Max-cut, and 94/93 for Max-2-SAT. The second part of this paper presents a “reverse” of the FGLSS connection by showing that an NP-hardness result for the approximation of Max clique to within a factor of N¹(g+1/) would imply a probabilistic verifier for NP with logarithmic randomness and amortized free-bit complexity g. We also show that “existing techniques” won´t yield proof systems of less than two bits in amortized free bit complexity. Finally, we initiate a comprehensive study of PCP and FPCP parameters, proving several triviality results and providing several useful transformations