Proof Checking and Approximation: Towards Tight Results

The last few years have seen much progress in proving “non-approximability results” for well-known NP-hard optimization problems. As we know, the breakthrough has come by the application of results from probabilistic proof checking. It is an area that seems to continue to surprise: since the connection was discovered in 1991 (Feige et. al. [21]), not only have non-approximability results emerged for a wide range of problems, but the factors shown hard steadily increase. Today, tight results are known for central problems like Max-Clique and Min-Set-Cover. (That is, the approximation algorithms we have for these problems can be shown to be the best possible). Such results also seem to be in sight for Chrom-Num. These are remarkable things, especially in the light of our knowledge of just five years ago. And meanwhile we continue to make progress on the Max-SNP front, where both the algorithms and the non-approximability results are improving, so perhaps there is more to come. This (short) survey will try to explain how the best known results are being derived. There is here a story that I believe is interesting and yet, although of course building on previous work, can be told in a a relatively self-contained way while still illustrating the central themes. It is characterized by the finding and exploitation of ever deepening “interactions” between proof checking and approximation, and the building of proof systems of unusual but powerful nature which not only have impressive consequences in approximation but may be of interest in their own right.