# Graph Classes A Survey Bibtex Bibliography

This paper continues the investigation of the connection between probabilistically checkable proofs (PCPs) the approximability of NP-optimization problems. The emphasis is on proving tight non-approximability results via consideration of measures like the ‘‘free bit complexity” and the ‘‘amortized free bit complexity” of proof systems.

The first part of the paper presents a collection of new proof systems based on a new errorcorrecting code called the long code. We provide a proof system which has amortized free bit complexity of \(2 + \epsilon\), implying that approximating Max Clique within \(N^{\frac{1}{3}-\epsilon}\), and approximating the Chromatic Number within \(N^{\frac{1}{5}-\epsilon}\), are hard assuming NP \(\neq\) coRP, for any \(\epsilon > 0\). We also derive the first explicit and reasonable constant hardness factors for Min Vertex Cover, Max2SAT, and Max Cut, and improve the hardness factor for Max3SAT. We note that our non-approximability factors for MaxSNP problems are appreciably close to the values known to be achievable by polynomial time algorithms. Finally we note a general approach to the derivation of strong non-approximability results under which the problem reduces to the construction of certain ‘‘gadgets.”

The increasing strength of non-approximability results obtained via the PCP connection motivates us to ask how far this can go, and whether PCPs are inherent in any way. The second part of the paper addresses this. The main result is a ‘‘reversal” of the FGLSS connection: where the latter had shown how to translate proof systems for NP into NP-hardness of approximation results for Max Clique, we show how any NP-hardness of approximation result for Max Clique yields a proof system for NP. Roughly our result says that for any constant \(f\) if Max Clique is NP-hard to approximate within N^{1/(1+f)} then \(NP \subseteq \overline{\text{FPCP}}\left[log, f \right]\), the latter being the class of languages possessing proofs of logarithmic randomness and amortized free bit complexity \(f\). This suggests that PCPs are inherent to obtaining non-approximability results. Furthermore the tight relation suggests that reducing the amortized free bit complexity is necessary for improving the non-approximability results for Max Clique.

The third part of our paper initiates a systematic investigation of the properties of PCP and FPCP as a function of the various parameters: randomness, query complexity, free bit complexity, amortized free bit complexity, proof size, etc. We are particularly interested in ‘‘triviality” results, which indicate which classes are not powerful enough to capture NP. We also distill the role of randomized reductions in this are

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