SETH vs Approximation

Our story is about hardness of problems in P, but its roots begin with two algorithmic approaches that have been developed to cope with NP-hard problems: approximation algorithms and fasterthan- brute-force algorithms. Approximation algorithms were proposed as a response for NP-hardness almost immediately (in historical perspective of almost half a century), and have been one of the most celebrated success stories of our eld. An outstanding complexity result in this area, which has since turned into a sub- eld of its own, is the Probabilistically Checkable Proof (PCP) Theorem. For many problems like Max-3-SAT we now have nearly tight hardness-of-approximation results.